
Class '^d Jc^ OC ^ 



Book._ 
CopyrightN^. 



1 €i 



/^/^ 



Ci)FffiIGHT DEPOSm 



WORM GEARING 



McGraw-Hill Dookfompaiiy 

Puj6tis/iers (^3oo£§/br 

ElGCtrical World TheEnginGGiin^ and Mining Journal 
tngiaeering Record Engineering Nows 

Kailwa/A^e GazettG American Machinist 

Signal LnginoGr American Engineei* 

Electric liailway Journal Coal Age 

Metallurgical and Chem ical Lngineering P o we r 



s»* -■■:S^'* 




Frontspiece. 

A Fifteenth Century Worm Gear from an Etching by Albrecht Diirer. 



WORM GEAMNG 



BY 
HUGH KERR THOMAS 

M. I. MECH. E., M. A. S. M. E., M. I. A. E. 



Second Edition 
Revised and Enlarged 



McGRAW-HILL BOOK COMPANY, Inc. 

239 WEST 39TH STREET. NEW YORK 



LONDON: HILL PUBLISHING CO., Ltd. 

6 & 8 BOUVERIE ST„ E. C. 

1916 



\ 



*Si t .» Vi ' .^ C^^ i"* 






Copyright, 1913, 1916, by the 
McGraw-Hill Book Company, Inc. 




m -7 1916 



THE MAPIiE PRESS X O R K PA 



'CI.A445532 



PREFACE TO SECOND EDITION 

A second edition of this book having been called for, the 
opportunity is afforded for making corrections of a few 
typographical errors, which appeared in the first edition. 
Part of Chapter IX has been rewritten, and three short 
appendices added to bring the text abreast of the writer^s 
most recent investigations. It is hoped that the value of 
the book is thereby considerably enhanced. 

H. K. T. 



PREFACE TO FIRST EDITION 

In the following work an attempt has been made, it is 
believed for the first time, to deal exhaustively with a little 
understood branch of applied mechanics. A complete analysis 
of the principles of the design of worm gearing has been made, 
and, primarily, this has been treated in its application to the 
rear axles of automobiles. For a number of years this type 
of gearing has enjoyed considerable popularity in Great Britain, 
and its wider use is daily becoming more general. 

From his experience with his own staff, as well as with many 
professional acquaintances, the author believes that the book 
will give information, which, although possibly possessed by a 
few engineers, has not hitherto been accessible to the designer 
and draughtsman in anything like a complete form. 

An important use for worm gearing is rapidly developing 
in connection with the reduction gear of steam turbines for 
marine propulsion; while these are, mechanically speaking, 
simpler problems, by reason of their greater stability, than the 
gears of an automobile, precisely the same rules can be applied, 
and the same thing may be said of gears for driving line shaft- 
ing from electric motors. 

vii 



viii PREFACE 

A brief outline of the work was given in two papers contrib- 
uted by the author to the "Automobile Engineer'^ for May and 
June, 1912, and some of the formulae there published, with 
much amplification, have been incorporated in the text. The 
subject is a complicated one, demanding lengthy explanations, 
but by confining the calculations to the use of elementary alge- 
bra and trigonometry, it is hoped that the solution of the vari- 
ous problems has been as far as possible simplified. 

The literature on the subject is extremely meagre; where 
it has been consulted, reference to the author has been given in 
the text; with very few exceptions, however, the work is entirely 
original, and the rules given have been in every case referred to 
practical experiments for verification. 

The author wishes to acknowledge many valuable sugges- 
tions in the progress of the work from his assistants, John 
Younger, B. Sc, Lewis P. Kalb, M. E., and C. P. Schwarz, 
D. Sc, to the former of whom is due the method for determin- 
ing the width of the worm wheel. 

He is also indebited to A. L. Cox for valuable assistance in 
reading the proofs and preparing the index. 

H. Kerb Thomas. 

London, January, 1913. 



CONTENTS 



Page 
Pkefacb V 

CHAPTER I 
Introductory 1 

CHAPTER II 
Choice of Materials and Methods of Manufacture , 5 

CHAPTER III 
Definitions and Symbols 8 

CHAPTER IV 

Preliminary Proportions 13 

CHAPTER V 
Pressure Angle and Form of Thread 22 

CHAPTER VI 
Strength of Worm Wheel Teeth • 34 

CHAPTER VII 

Stresses in Worm Gearing 37 

CHAPTER VIII 
The Width of the Worm-wheel 47 

CHAPTER IX 
The Temperature Coefficient 58 

CHAPTER X 
Efficiency of Worm Gearing 63 

CHAPTER XI 
General Points of Design of Mounting 76 

CHAPTER XII 
Recapitulation of Formula Used 81 



IX 



X CONTENTS 

APPENDIX 

A. Alternative Method of Calculating Stresses in Worm 

Gearing. . 85 

B. Efficiency of Worm Gearing 89 

C. Reversibility. 91 

Index 95 



WORM GEARS 

CHAPTER I 

INTRODUCTORY. 

Worm, or more properly screw, gearing is of great antiquity; 
the word screw, directly derived from the Danish scrue, is defined 
in the Encyclopaedia Britannica (11th edition) as "si cylindrical 
or conical piece of wood or metal having a groove running 
spirally round it." Such a spiral was first studied geometrically 
by Archimedes (287-212 b. c.) and described in his work Uepl 
€\tK0)v, which deals in 28 propositions with various mathe- 
matical problems arising out of the construction of a helix. Un- 
fortunately but too many inventions of the early engineers have 
passed out of human knowledge, and the earliest form of worm 
gearing in which an Archimedian spiral was employed to rotate 
a toothed wheel is not now on record. We have, however, a 
series of curious drawings by the German artist, Albrecht Diirer 
(1471-1528), which were engraved on wood by the Master, to 
the order of the Emperor Maximillian. It is on record that His 
Majesty, conceiving for his own aggrandizement a Triumphal 
Procession, commissioned his favorite painter to prepare 
designs of the emblematic cars which were intended to figure 
therein. A number of these designs are still in existence and 
one is reproduced here in the frontispiece. It will be seen that 
the vehicle, which is of heroic proportions, is propelled by all 
four wheels, each actuated by a perfect worm gear. It is thus 
a "four wheel driven'' car in the most modern sense of the 
word and is at the same time undoubtedly the earliest worm 
driven vehicle of which any record exists. Unfortunately, we 
can hardly believe that so ponderous a machine could have 
ever been propelled by means of such obviously inefficient 
gearing operated by only "four man power." It is not known 

1 



2 WORM GEARS 

whether this remarkable car was ever made, but it is certain 
from the existing prints that Dtirer must have seen, somewhere 
or other, a worm and wheel from which his elaborate design 
was copied. 

From the fifteenth to the nineteenth century, almost all 
gearing was made of wood, and in old treatises on gearing, 
numerous examples of wooden worm gears are illustrated. 
Several examples of worm-driven traction engines are to be 
found in the records of the British Patent Office, and more than 
one are for the propulsion of tram cars, but as we know it to-day, 
w^orm gear, which had been used for many years in various 
forms, was first applied to the purpose of driving the road 
wheels of automobiles by Mr. F. W. Lanchester, who, employ- 
ing it in his first cars designed and built before the close of the 
nineteenth century, has consistently used it ever since. The 
Brothers Dennis adopted it for 3 1/2-ton commercial vehicles 
and subsequently for pleasure cars a few years later, and for 
several years the Lanchester Company and the Dennis Com- 
pany were the only firms regularly using it. Much opposition 
to its general adoption existed for years but Lanchester and 
Dennis holding confidently to their belief in the principle, it 
began to be realized in England that worm gearing for final 
drives of automobiles was a serious competitor of bevel gears 
and could be considered as having passed the experimental 
stage. Gradually other builders took it up and made it success- 
ful in wider use until finally the seal of approval was set by the 
enormous fleet of vehicles built and owned by the London 
General Omnibus Company, all driven through this type of 
gear. 

A few attempts, distinguished by lack of self-confidence, were 
made to use it in the United States, with little success on the 
part of manufacturers or enthusiasm shown by the public, 
until the author, who had experimented with it for two years 
in England, was privileged to introduce it into the United 
States, in the year 1911, on an extensive scale, as part of the 
regular product of The Pierce- Arrow Motor Car Company, and 
its use is now very general in England, on the Continent of 
Europe, and is rapidly extending in America. 



INTRODUCTORY 3 

Certain obvious advantages have contributed to its adoption 
in automobiles of all kinds from the lightest pleasure cars to the 
heaviest commercial vehicles, and, while there is everything to 
be said in its favor, its design has been surrounded with so 
much mystery that many have been deterred from adopting it. 
It is true that some writers have attempted to ventilate the 
subject, and some mention of it will be found in most works 
devoted to gearing, but the author has searched in vain for any 
book treating the whole subject comprehensively. Literature 
specially devoted to worm gears for automobiles is, indeed, 
practically non-existent, and, at the best, consists of occasional 
contributions to technical magazines. Most of such articles 
omit altogether the fundamental principles, a careful study 
of which is a necessity if successful result is to be obtained by 
other methods than mere chance. 

Much opposition to the use of worm gearing has been due to 
a misconception, for which practically all the text-books are 
responsible, that it is inefficient, and (for automobile work) 
irreversible; it will presently be shown that it is neither, but 
mention may be made here of a remarkable kind of gear which 
appeared in the closing years of the nineteenth century. It 
was known as the ''Globoid Gear" and the supposed excessive 
friction of an ordinary worm was sought to be overcome by an 
arrangement of conical rollers mounted upon radial pins 
around the periphery of the worm wheel; these rollers were 
meshed by a worm of very large pitch and the gear certainly 
worked freely in both directions; beyond that, it was very 
cumbersome, and after a little use noisy. It was never used in 
automobiles, and is only referred to here to show how the action 
of this type of gearing has been misrepresented — even to the 
extent of inducing inventors to develop complicated mechanisms 
to overcome supposed faults which do not in reality exist. 

Since the early days of experiments so much actual experience 
in daily use with worm gears of all sizes has been possible that 
it can now be said that something approaching finality has 
been reached, and the art of making worm gearing has been 
placed upon a substantial foundation. 

Such is, in outline, the history of this most interesting me- 



4 WORM GEARS 

chanical device, and with this very brief glance at its early use, 
we will pass to the consideration of the principles of its design 
and construction, dealing in as progressive a manner as possible 
with a subject which is necessarily of a somewhat complex 
nature. 



CHAPTER II 

CHOICE OF MATERIALS AND METHODS OF 
MANUFACTURE 

At different times, many combinations of metals have been 
proposed for making worm gears. Some kind of bronze for the 
wheel and steel for the worm was adopted as the most natural 
combination for two sliding sm'faces working together; the first 
and most obvious improvement being to case-harden the worm 
to minimize wear. Rise in the cost of copper and tin suggested 
the use of other metals for the wheel, and cast iron of various 
grades, mild steel, and even case-hardened steel were tried for 
the wheels, all in combinations with and without a hardened 
worm. For many years, bronze has been used for the worm 
wheels of elevators driven by a steel worm and it is now gen- 
erally recognized that case-hardened steel for the worm and 
phosphor bronze for the wheel cannot be surpassed. It is 
interesting to note here that manganese bronze is unsuitable, 
presumably on account of its hardness. The bronze must be 
very homogeneous and close grained, and great care must be 
taken to ensure uniformity of mixing, melting and pouring, in 
order to ensure uniform results. The writer recommends a 
mixture of the following proportions: 

Copper 89 

Phosphorous 1 

Tin 10 

This will have an elastic resistance to crushing amounting to 
22,000 lb. per square inch, and to penetration, under a hardened 
steel point of .125 sq. in. area, of 5000 lb. On the Shore 
scleroscope, it will give a hardness reading of 15 to 20. Its 
tensile strength will amount to 35,000 lb. per square inch. 
With such a metal, a safe working stress of 7000 lb. per square 
inch may be permitted 

5 



6 WORM GEARS 

For the worm, seeing that its shape conduces to great strength, 
no unusual strength is required in the material, and almost any 
low carbon case-hardening steel, which can be heat treated 
without serious distortion, may be employed. Steel for this 
purpose should contain 3 to 3.5 per cent, nickel and .16 to .18 
carbon. In its untreated state, it will give approximately the 
following physical tests, on a standard 2-in. specimen. 

Elastic limit 62,000 lb. per square inch. 

Maximum strength 82,000 lb. per square inch. 

Elongation 30 per cent. 

Reduction 60 per cent. 

After machining to size with an allowance of say .003 in. 
for grinding, it can be subjected to the following heat treat- 
ment. Pack in carbonizing material and maintain for eight 
hours at a temperature of 1600 to 1650° F., allow to cool in the 
carbon, reheat to 1350° F. and quench in oil. 

The above may, of course, be varied at the discretion of the 
designer, the object being to obtain a thick casing of a hardness 
which will indicate from 60 to 70 on the scleroscope. 

After hardening, the worm may be finished to required size 
by grinding which will, if properly performed in a suitable 
machine, provide a worm accurate upon all threads and with a 
perfect and lasting working surface. 

With regard to the method of manufacture of gears, the cast- 
ing of the wheel should be rough turned and roughly gashed in 
a milling machine, after which it is strongly recommended that 
it be set on one side to season. Being necessarily of consider- 
able mass, the surface tension set up during cooling, on being 
released by machining will cause some distortion, and seasoning 
for a period of some weeks will allow the metal to assume a 
permanent form which it may be relied upon to maintain. The 
wheel is then turned to exact size and the teeth cut by means of 
hobbing. It is safe to say that few makers have, until recently, 
papreciated the necessity for employing extra rigid machinery 
for this purpose without which the highest class of work is 
impossible. By the older method of hobbing, a parallel hob 
was fed radially into the wheel, but in more modern methods 
a taper hob is fed at fixed centers tangentially to the wheel it 



CHOICE OF MATERIALS 7 

is cutting; the reason for this is obvious — by such a method 
alone can the centers be preserved at the required distance and 
interchangeabihty of similar gears be assured. It is not possible 
to obtain the same finish on the worm wheel as in the case of the 
worm, but some makers have been particularly successful in 
this respect, finishing the worm wheel with a special kind of hob 
which leaves an extremely accurate surface which immediately 
beds to the worm. 

In the case of the worm, it is an advantage to rough turn it 
and rough cut the thread, and then anneal it, to remove internal 
strains and subsequent distortion, and while in some steels it is 
not necessary, it is a practice which may be recommended. The 
cutting of the worm may be performed in a lathe with a single 
point tool accurately ground to the required form, and for 
experimental work, saving the expense of cutters, this has its 
uses; the better method, however, is of course, to mill the thread 
in a thread milling machine, and special machines are generally 
used for repetition work. In the case of the hollow or Hindley 
worm, as used by Mr. Lanchester and others who have followed 
his example, a special bobbing machine is used for cutting the 
worm, to which reference will be made again later. Such a 
worm cannot be ground, and, after hardening, must be cleaned 
up on its working surfaces by hand. At the time of writing, 
the author is not aware of any parallel worms having been 
hobbed, but it seems likely that there may be developments in 
this direction in the future. Whatever method is adopted, 
however, the greatest possible accuracy must be insisted upon 
in the case of both the worm and the wheel, the high efficiency 
of modern worm gears being almost entirely due to extreme care 
in this respect. 



% 



CHAPTER III 

DEFINITIONS AND SYMBOLS 

A worm gear may be defined as a spur wheel which is rotated 
by an endless rack, the teeth of which are successively pressed 
against the teeth of the wheel. By making the rack teeth in 
the form of a spiral and rotating it upon its axis (sloping the 
wheel teeth to a corresponding angle), the effect of an infinitely 
long rack is obtained. 

Such a rack is called a parallel worm. By revolving the worm 
wheel, the teeth of the rack may be caused to move along, that 
is to say, the worm will itself commence to rotate, the relative 
motions being thus convertible. The worm is then said to be 
'' reversible, '^ the amount of reversibility depending upon 
certain fixed principles which will be later discussed. 

In order to obtain a greater area of surface contact between 
the wheel and the rack, the latter may be curved to partially 
embrace the wheel, the form of the worm then partaking of 
that of an hour glass. This form is usually known as the Hind- 
ley worm, from its inventor, and has been brought to a high 
state of perfection for use in automobiles by Mr. Lanchester. 
As the central portion of a Hindley worm exactly corresponds 
in its action with that of a parallel worm, the same calculations 
are applicable to either, with the exception of such formulae as 
refer to the number of teeth in engagement at one time and 
consequently the length of the worm; these differences will be 
treated in their proper place. 

We will now define clearly the following expressions to which 
frequent allusion will have to be made. 

Gear Ratio. — As in any other system of gearing, the ratio 
of the speeds of revolution which one member will make when 
meshing with the other is called the gear ratio; thus, with a 3 to 

8 



DEFINITIONS AND SYMBOLS 9 

1 ratio, it is to be understood that the worm will make exactly 
three revolutions to one revolution of the worm wheel. Unlike 
spur or bevel gears, however, the ratio is not primarily depend- 
ent on the number of threads of the worm, but involves the 
question of lead. 

Lead. — ^This may be defined as the distance traveled along 
the axis by one thread in one revolution; it is measured in 
inches. How this determines the gear ratio may be made clear 
by the following example. If the worm wheel be 30 in. in 
circumference and the worm has a lead of 6 in., the gear ratio 

30 
will be -^ = 5 to 1. This may be further defined as the dimen- 
sional ratio, obtained by dividing the circumference of the wheel 
by the lead, so that we may say as a starting-point that 

wheel circumference ,. ,^. 

-, — , = gear ratio (1) 

worm lead 

By increasing the diameter of the wheel the number of teeth 
is increased and the gear ratio is reduced, while the converse 
of this is, of course, true, but by increasing the diameter of the 
worm the rubbing velocity is increased. 

By increasing the lead angle, the gear ratio is reduced; the 
circumferential pitch being first determined and the number of 
wheel teeth, it follows that a single-threaded worm will give a 
reduction of 1 to n where n = the number of wheel teeth. A 
double-threaded worm will have a reduction oi2ton and so on, 
but this relationship is not primarily due to the number of 
threads in the worm but to the lead of the thread which may be 
extended to include at each revolution one, two, or more teeth. 
Thus, in a worm having five threads, each thread will make a 
complete engagement with only every fifth tooth in the wheel, 
and the other four threads will pick up the intervening teeth 
respectively. 

Pitch Line of WheeL — ^Properly speaking a worm wheel has 
no pitch line at all, that is to say, in the sense of the two rolling 
circles of spur or bevel gears; it is, however, a convenient expres- 
sion for a circle corresponding to the mean effective diameter of 
the wheel and must be so understood. 



10 WORM GEARS 

Pitch Line of "Worm. — This also has only an hypothetical 
existence, but may be defined as the circle described about the 
axis of the worm which touches, at right angles, the pitch circle 
of the wheel. All subsequent references to '^Pitch" whether 
applied to the worm or wheel must be understood to be meas- 
ured upon these circles. 

Circular Pitch of Wheel. — The distance in inches, or fractions 
of an inch, between the centers of the wheel teeth measured 
along the pitch line. 

Axial Pitch of Worm. — The distance in inches, or fractions of 
an inch, between the centers of adjacent worm threads, meas- 
ured along the pitch line, parallel to the axis of the worm. 

Note. — In a worm having more than one thread, this is not 
the same as the lead of the worm. In fact, the lead is the prod- 
uct of the number of threads or starts multiplied by the 
axial pitch. The axial pitch of the worm is always equal to the 
circular pitch of the wheel. 

Normal Pitch of Worm and Wheel. — This is the distance 
between the centers of two adjacent threads or teeth measured 
normally or at right angles to their faces along the pitch line. 
Owing to the inclination of the threads or teeth, this distance 
is always less than the circular or the axial pitch. 

Circimiferential Pitch of Worm. — The distance along the 
pitch line between two adjacent worm threads, measured cir- 
cumferentially round the worm. 

Lead Angle. — The angle formed by the thread of the worm 
with a line drawn at right angles to the axis of the worm. If 
the spiral formed by the threads were unwrapped from the 
worm, it would form an inclined plane having an inclination 
equal to the lead angle. 

Addendimi. — The height of wheel tooth or worm thread 
outside pitch line. 

Dedendtmi. — The depth of wheel tooth or worm thread inside 
pitch line. 

Length of Worm. — ^The length of the cylinder bounded by the 
pitch line of the worm. If two radii be drawn from the center 
to the circumference of the pitch line of the wheel, so that each 
touches the extremity of this cylinder, they will make an angle 



DEFINITIONS AND SYMBOLS 11 

which is subtended by the cyHnder of the worm pitch Kne, and 
the length of this cyhnder is the chord of this angle. 

Note. — The length of the worm has nothing to do with the 
the lead of the worm. 

Subtended Angle of Worm. — If two radii be drawn from the 
axis to the circumference of the pitch circle of the worm, they 
will form an angle which is filled by the teeth of the worm 
wheel. This is the subtended angle of the worm, and is the only 
manner in which the effective width of the worm wheel can be 
measured. The determination of this angle has a very impor- 
tant bearing upon the performance of the gear. 

Included Angle of Thread. — In a V-threaded worm, the angle 
made by the inclination of the faces of the V is the included 
angle. 

Pressure Angle.— Half the included angle. 

Note. — Both the included angle and the pressure angle may 
be measured normally or in the direction of the axis; unless 
otherwise specified, however, the axial angles are always to be 
understood to be implied. 

The following system of symbols will be employed through- 
out this work. 

G =Gear ratio. 

D =Pitch line diameter of worm wheel in inches. 

d = pitch line diameter of worm in inches. 

L =Lead of worm in inches. 

I = Length of thread per revolution in inches. 

N = Number of teeth in worm wheel. 

n = Number of teeth in worm. 

R = Pitch line radius of worm wheel in inches. 

r = Pitch line radius of worm in inches. 

R^ == Radius of road wheel in inches. 

i^" = Extreme radius of worm wheel on plane of worm axis. 

V = Rubbing velocity in feet per second. 

T = Torque in pounds at pitch line of worm. 

p = Tangential pressure at pitch line in pounds. 

p' = Normal tooth pressure. 

P = Circular pitch of worm wheel and axial pitch of worm. 



12 WORM GEARS 

P' = Normal pitch of worm threads and wheel teeth. 

P'' = Circumferential pitch of worm threads. 

/ = Force separating worm and wheel. 

w = Weight at rear axle in pounds on the ground. 

X =Safe load on tooth of worm wheel, in pounds. 

g = Length of worm in inches. 

n' = Number of teeth in contact. 

a =Lead angle. 

= Included angle of worm thread (axial). 

W = Normal included angle of thread. 

o = Stress permissible in wheel teeth in pounds per square 

inch. 

;- = Gliding angle. 

f = Angle of friction (tan <p = n). 

pL = Coefficient of friction. 

T] = Mechanical efficiency. 

/? = Angle of worm subtended by wheel. 

CO = Pressure angle = - * 

X = Length of tooth. 

P 

z = Thickness of worm thread at pitch line = ~* 

K = Dedendum. 

= Addendum. 

p = Revolutions of worm per minute. 

d = Angle of worm wheel subtended by worm. 



CHAPTER IV 

PRELIMINARY PROPORTIONS 

Certain data are required before the design can be attempted. 
It is usually necessary to know the following: 

(a) The gear ratio. 

(b) The maximum torque of the motor. 

(c) The lowest gear ratio in the transmission. 

Note. — ^The torque delivered to the worm will be the maxi- 
mum torque of the motor multiplied by the lowest gear ratio 
(generally the reverse gear) less the drop in mechanical efficiency 
due to the gears and bearings. 

(d) The maximum speed of the motor and especially the 
speed at which the maximum torque is given off. In an internal 
combustion motor the speed of maximum torque is always less 
than the speed of maximum power. 

(e) The total weight on the ground at the rear axle. 

(f) The diameter of the road wheels. 

The first thing to settle is the circular pitch, and while a great 
deal of latitude is permissible in this, some limitations must be 
set to it. It will be shown that the strength of the wheel teeth 
varies directly with the circular pitch; the smallest pitch even 
in the lightest cars which can be employed is 13/16 in., since 
less than this gives very little margin for wear or for hidden 
defects in the bronze of the worm wheel, but it is clearly im- 
possible to increase the circular pitch in direct proportion to 
the power to be transmitted. The pitch diameter of the worm 
naturally affects the torque loads on the teeth, and it will be 
necessary to increase the worm diameter in some proportion to 
the power, so that it seems that some increase of the circular 
pitch with the diameter of the worm is a natural sequence. 
Very coarse pitches are in use notably in the gears adopted as 
standard in the vehicles of the London General Omnibus 

13 



14 WORM GEARS 

Company which are in the neighborhood of 1 7/8 in. but there 
is no advantage in going above 1 1/4 in. The author has 
employed 1 1/8 in. in gears transmitting over 90 H.P. through 
a worm with a pitch line diameter of no less than 4 3/4 in. This 
in heavy pleasure cars has given perfectly satisfactory results. 

The disadvantage of a coarse pitch is this, when at work, the 
surfaces of the threads and the teeth are separated by a film of 
oil, the coarser the pitch, the fewer the number of teeth in con- 
tact, consequently the tangential pressure between the worm 
and the wheel must be carried on fewer films of oil, that is to 
say the specific tooth pressure will be greater and this may rise 
to a point where the oil is squeezed out and lubrication becomes 
ineffective; a fine pitch is therefore always better than a coarse 
one, other things being equal. 

There is another aspect to this, the worm must always be 
rigid beyond the possibility of springing between its journals, 
with a very coarse pitch so much stock may be removed from 
the body of metal as to seriously reduce its transverse strength, 
moreover some slight angular torsion must theoretically occur 
in the worm itself, and if the cross-section of the threads is 
considerably greater than that of the central spindle, there is a 
tendency for the threads to unwrap and tear away from the 
body of the structure; a similar result being very familiar to 
automobile engineers in the case of heavily stressed splined 
shafts, which, if the splines be very coarse and rigid in com- 
parison with the rest of the shaft, will tear asunder like the 
quarters of an orange. 

All these considerations therefore point to the desirability of 
employing a comparatively fine pitch, and in order to ensure 
stiffness in the spindle and to leave sufficient metal in this, a 
pitch diameter less than 2 in. is seldom required. 

Based upon examples from actual practice, the following 
pitches as shown in Table 1, will be found satisfactory for 
different horse-powers : 

These are shown diagrammatically in Fig. 1, and while it 
may be expedient to depart slightly from these relationships to 
suit special features of axle design, they should be as closely as 
possible adhered to. 



PRELIMINARY PROPORTIONS 



15 





TABLE I 




II.P. 


Circ. pitch 


Pitch dia. 


15 


i|in. 


2iin. 


25 


1 in. 


2f in. 


50 


1 in. 


3iin. 


75 


11 in. 


4iin. 


100 


li in. 


5 in. 



5.5 

5.0 
4.5 
4.0' 
3.5 
3.0 
2.5 
2.0 
1.5 
1.0 
0.5 























^ 


















^ 


^ 
















,^4 


y^ 


^ 














v,^ 




t^ 
















^ 


-jj^ 
















^ 








































































CircuU 


X Pitcli 


_of_^ 


ee} 







— 





































































10 20 30 40 50 60 70 80 90 100 

Horse Power 

Fig. 1. 

Having selected from this table the suitable pitch and worm 
diameter for a given horse-power, the next thing to determine 
is the number of wheel teeth and worm threads to give the 



16 



WORM GEARS 



required ratio. Seeing that a comparatively small pitch has 
been selected for all wheels, it is evident that a relatively greater 
number of worm threads is possible. 

DETERMINATION OF WORM WHEEL DIAMETER 

This will naturally be preferred as small as possible but it 
must not be reduced at the expense of excessive tangential 
pressure, and the number of teeth should not be less than 24 or 
more than 40. The writer has found that almost any conditions 
can be satisfied with between 30 and 40 teeth, bearing in mind 
that a very heavy vehicle requires a larger wheel than a light 
one, but the heaviest commercial vehicle need not have more 
than 40, and questions of road clearances under the axle will to 
a great extent determine the diameter which must always 
remain a matter of judgment on the part of the designer. 

The following proportions of teeth are recommended: 




Fig. 2. 



Addendum, 
Dedendum, 



.3183 P 
.3683 P 



It has been stated that the pitch of the worm can be measured 
in three ways (Fig. 2) : 

Axially =P 

Normally =P' 

Circumf erentially = P". 

and these are inter-related as follows: 



PRELIMINARY PROPORTIONS 17 



P' = P cos a 
P" = P' cosec a 
Therefore, P" = P cos a cosec a 

P" = P cot a 



(2) 



It follows from this that the number of threads on a worm 
has the following relation to its pitch line diameter. 

nd = nP cot a (3) 

Tzd 

.'. n = ^ — .— (4) 

P cot a ^ ^ 

and to find the number of threads, it is first necessary to know 
the value of the lead angle a which again depends upon the 
lead L, and this in its turn upon the gear ratio required. 
The gear ratio G has the following value 

ttD 



^-L 


(5) 


hence ,. ^D 

^~ G 


(6) 


and the lead angle a is such that 




tan a— , 
m 


(7) 


Substituting from equation (6) 




nD 




G 

tan « = — T 
nd 




7:0 




D 

tan a- ^^ 


(8) 


from which a is at once found, and from 


this the value of n 


(equation 4). 





DETERMINATION OF LENGTH OF WORM AND NUMBER OF 

TEETH IN CONTACT 

The worm and worm wheel may be regarded as two cylinders 
which cut one another, their axes being at right angles. 



18 



WORM GEARS 



In Fig. 3, which is a section through the axis of the worm, the 
Hne dc represents the hmiting surface of the worm; o is the 
addendum of the teeth. 

R'^, the radius of the hmiting circle of the worm wheel, is 
equal to R+o. 




Fig. 3. 

The total height of the wheel teeth cutting the cylinder of the 
worm is +0, and this is also the versin of the angle-. 



Therefore, 
Again, 



2X0 



b 



R- = ^^^^^^ 2 



dc = 2 sin- 



d 



,'. the length oi dc = 2 R" sin ~=g 



(9) 



Since the teeth of the wheel are hobbed, it follows that every 
space between the worm threads has a wheel tooth corre- 
sponding to it and some portion of this is in contact with a 
thread of the worm. 



The number of threads is ^ 



(10) 



PRELIMINARY PROPORTIONS 19 

and this is the number of teeth in contact at all times. Let 
this =n\ 

Then, n'=| (11) 

Substituting for g, 

2R" sin^ 
2 (12) 



n 



P 

Up to this point we have considered the method of determin- 
ing the following: 

Pitch of the wheel teeth, 

Diameter of worm, 

Diameter of worm wheel. 

Lead angle, 

Number of worm threads. 

Length of worm. 
The centers of the worm and wheel are evidently expressed 
by the following 

c = — 2— (13) 

and having thus settled the leading dimensions, we are in a 
position to investigate the correct form of thread, and the 
required strength of the various parts to withstand the stresses 
to which they will be subjected. 

It is convenient at this point to introduce a number of 
formulae which will frequently have to be referred to in the rest 
of this investigation. 

Lead of worm L = Pn (14) 

Length of thread per revolution =l = 7:d sec a (15) 

hence, rubbing velocity of worm 
TrdsecaXr.p.m. 



12X60 



ft. per second. 



;rc?seca:Xr.p.m. .^^. 

"= 720^ ^'^^ 



Lead angle a may be found from 



tan a = —y- (17) 



20 WORM GEARS 

Whence it follows that the angle of lead is not affected by the 
number of teeth in the wheel. 

The rubbing velocity v is given by some writers as being the 
circumferential speed of the worm; this is incorrect and would 
only be true of a worm having its lead angle =o; it is of course 
dependent upon the length of thread (per revolution of worm) 
measured around the spiral and equation (15) gives the correct 
value. The manner in which the rubbing velocity is affected 
by the diameter of the worm may be illustrated in the follow- 
ing manner. 

It is evident that a given lead may be obtained by a large 
diameter worm with a small lead angle, or a small worm with a 
very quick angle, but that similar results will give very different 
conditions of working is apparent from the following examples. 
Take the case of three worms having each a linear pitch of 5 in., 
but with pitch line diameters of (a) 3 in., (6) 5 in., and (c) 7 in. 
respectively. 

It follows from equation (17) that the lead angles of such 
worms will be as follows: 

5 5 

(a) tan a=^ = FrT^='531 whence a=2S° 

5 5 

(b) tan q:=^= -,c -^ =.318 whence a=17°39' 
^ on 15.70 

(c) tan q:=^= 2pgg=. 2274 whence a= 12° 49' 

and the rubbing velocities from equation (16) will be at 1000 
revolutions per minute. 

^.1.132X9.42X1000 ^ . _ „^ , 

(a) ^^7j =14.8 it. per second. 

,,, 1.049X15.7X1000 ^^ ^ ,, . 

(b) ^^7T = 22.9 ft. per second. 

, ^ 1.025X21.99X1000 ^ . _ ,^ . 

(c) j^ =31.3 it. per second. 

The immediate effect of this is at once apparent, since to 
revolve the worm wheel for one second of time the surfaces of 



PRELIMINARY PROPORTIONS 21 

the three worms referred to will have to travel through these 
distances in order each to revolve the worm wheel through the 
same amount and, as the lost work of the worm per revolution 
is represented hylXiiXp' where I is length of thread per revolu- 
tion, fji is coefficient of friction, and p^ the tooth pressure, it 
follows that the losses will be directly proportional to the length 
of thread, that is, the rubbing speed. 



CHAPTER V 

PRESSURE ANGLE AND FORM OF THREAD 

It has already been remarked that much misconception 
formerly existed as to the possibility of any worm gear being' 
reversible, that is to say, that an automobile so fitted would be 
unable to ''coast." In this connection, the form of thread is 
of the utmost importance and by its manipulation, we can 
provide within practical limits perfect reversibility with any 
lead angle of ordinary dimensions. 

Let it always be remembered that a section through the worm 
on a plane including its axis, will represent it as a rack, and the 
worm wheel as a pinion rolling upon it. It is not necessary to 
state that if the sides of the teeth of the rack be straight, the 
teeth of the pinion will be of an involute form; moreover, if the 
wheel were of very large diameter, the sides of the teeth of the 
rack could be square or nearly so. The size of the wheel, how- 
ever, in an automobile is necessarily very limited, and conse- 
quently if the rack teeth or worm threads were made square, a 
great amount of undercutting would ensue in the generation of 
the wheel teeth, and to avoid this, the threads of the worm 
must be cut away in the form of a V, and this brings us to the 
question of the pressure angle. 

In their ''Practical Treatise on Gearing," Messrs. Brown & 
Sharpe recommend a pressure angle of 14 1/2 degrees, that is 
to say, the threads of the teeth will have an included angle of 
29 degrees; it is, however, pointed out in the same work, that 
interference of the rack teeth will begin in wheels of 31 teeth if 
this angle be adopted. Something larger than this will there- 
fore have to be selected. There is, however, another aspect to 
this, and in it is involved the question of reversibility. 

It must be remembered that the surface of the worm thread 
is constantly slipping over the faces of the wheel teeth, and the 

22 



PRESSURE ANGLE AND FORM OF THREAD 23 

direction in which this sHpping takes place is along a plane 
mutually tangent to the face of the thread, and the face of the 
wheel teeth. It is evident that if this gliding angle be 45 
degrees, it will be immaterial whether the worm drives the wheel 
or the wheel the worm; with a square thread, however, such a 
gliding angle is only possible if the proportions of the worm are 
such that they are satisfied by the equation 



t: 



d = L 



(18) 



The gear ratio and the dimensions of the worm and wheel may, 
and generally will, make this impossible. 

In Fig. 4, let a 6 c be a triangle where a c equals the lead and 
6 c is equal to the circumference of the pitch line of the worm. 
Then the angle a b c equals the lead angle of the worm. The 




working surface of the thread is shown in perspective by the 
plane a e dh and, in a square-threaded worm, the angle a b cis 
the gliding angle of the rubbing surfaces. As drawn, however, 
this angle is much less than 45 degrees and such a gear cannot 
be reversible, except to a slight degree. Now suppose the plane 
a e d b partially rotated about the line a 6 as an axis until it 
assumes the position a g f b, the angle which the plane makes 
with the horizontal plane db ch may become 45 degrees without 
any increment of the angle a b c, or in other words converting 
the square into a V thread, the gliding angle has been raised to 
45 degrees without increasing the lead angle. If then a gliding 
angle of 45 degrees were the one object to be attained, this 

would be satisfied by the equation a + ^ =45 degrees where a 



24 



WORM GEARS 



equals lead angle and 6 equals the angle included between the 
faces of the V thread as shown in Fig. 5; and any gear designed 




Fig. 5. 



upon these principles will be perfectly reversible, provided the 
angle of the pitch is kept within reasonable limits. 



TABLE II.— SHOWING REQUIRED ANGLE 6 TO GIVE A GLIDING ANGLE 

OF 45 DEGREES 



Lead angle, a 


Included angle, 6 


25 


40 


26 


38 


27 


36 


28 


34 


29 


32 


30 


30 


31 


28 


32 


26 


33 


24 


34 


22 


35 


20 


36 


18 


37 


16 


38 


14 


39 


12 


40 


10 


41 


8 


42 


6 


43 


4 


44 


2 


45 






PRESSURE ANGLE AND FORM OF THREAD 25 

Perfect reversibility will be secured if the above proportions 
are followed; it has, however, already been shown that a thread 
which is square or nearly so will cause destructive undercutting 
in the wheel teeth. How then in the case of a lead angle of 
45 degrees is reversibility to be obtained? 

It happens that with gears machined by modern processes, 
the coefficient of friction is very low; in recent experiments 
made upon the worm gears for operating the lock gates at the 
Panama Canal the coefficient of friction was so low that gears 
which were actually required to be irreversible were found to 
be the reverse in actual practice, and this fact may be taken 
advantage of by the designer of automobile gears inasmuch as 
the angles given in the above table may be departed from 
within very wide limits without in any way impairing the 
reversibility. 

With a narrow tooth, such as is given with a 14 1/2 degree 
pressiu*e angle, the difficulty of milling the worm is much 
increased, in cases where, as is usual, the lead angle is above 
28 degrees. By making a flatter tooth, that is one with a 
more open V, this difficulty to a great extent disappears, and 
the grinding of the worm teeth is also a far simpler operation. 

The author^s practice is to adopt for the sake of uniformity a 
universal angle of 60 degrees which greatly simplifies designing 
and which gives entirely satisfactory results in both manu- 
facture and operation, this angle being, of course, measured in 
a plane parallel to the axis of the worm. 

Throughout what follows then, a value of 60 degrees will be 
assumed for 6. 

The relation between the axial included angle and the 
included angle measured on a section cut normally to the worm 
thread is such that 

tan 2 

tan^ = ^ (19) 

2 cos a 



Hence 



¥ 6 

tan — = tan - cos a (20) 



26 



WORM GEARS 



and the milling cutters with which the threads of the worm 
are milled must be ground to this angle, ¥. 

Thirty degrees will consequently be the axial pressure angle 
recommended for general practice. 

It will be objected that the above practice is contradictory to 
the theory of reversibility — why use a gliding angle so much 
higher than the theoretical value? The answer is that experi- 
ence shows that reversibility is not affected nor is the efficiency, 
and a great advantage accrues from the absence of any under- 
cutting of the worm-wheel teeth, and the buttress form which 
is given them greatly increases their resistance to shearing and 
diminishes interference. 




Axial Section Through Worm Teeth 

Fig. 6. 



Fig. 6 illustrates a section cut through the worm in the direc- 
tion of the axis and fixes the proportions of the thread, all of 
which are based upon the pitch P, which is of course the same 
as the circular pitch of the wheel. The points of the teeth are 
shown with sharp corners, but in practice, when finishing the 
hardened worm, the extreme corners should be very slightly 
removed by grinding. In making the hob for cutting the worm 
wheel, the addendum o must be lengthened as shown by the 
dotted lines until it is equal to k, by which means a working 
clearance will be provided at the bottom of the teeth. This 
will give to the wheel teeth a dedendum equal to that of the 



PRESSURE ANGLE AND FORM OF THREAD 27 

worm K, and the height of the teeth in contact will be twice the 
addendum o or .6366 P. 

It must always be borne in mind that the pressure between 
the worm threads and the wheel teeth is transmitted along a 
line normal to a plane drawn mutually tangent to the teeth of 
the wheel and the threads of the worm at the momentary point 
of contact. When the worm is rotating, a number of these 
lines of pressure envelop the worm in a segment of a cone, whose 
apex is located in a prolongation of the worm axis, and whose 
base is the sector of the circle of the worm pitch line contained 
within the angle subtended by the worm wheel. Hence the 
thrust of the worm is not all taken up on the thrust bearing 
but is partly carried on the journal bearings in which the worm 
is mounted. This will be further considered when dealing with 
the stresses in bearings. 

The next point of consideration is the width of the wheel; 
this affects two things, first the length of the wheel teeth, and 
secondly the number of teeth in contact with the worm. 

On the length of the teeth, to a great extent, depends the 
strength of the worm wheel considered as a spur wheel and this 
is determined as follows : 




Fig. 7. 



The tooth pressures will, on reflection, be seen to be due to 
one of two causes. 

(a) The torque of the motor applied at the worm pitch line. 

(b) The negative tractive effort due to the momentum of the 
car when brakes are applied to the transmission. 

Dealing with (a), in Fig. 7, the developed thread surface is 



28 



WORM GEARS 



shown by the line AB, the developed pitch line by the line AC; 
T is the torque available at the pitch line. Then the weight w[ 
which can be lifted by T, will be expressed by the equation: 

T 

(21) 



W= 



tan (a: + 0) 

Where (/> is the angle of friction, the reaction between the 
surfaces at p' will then be represented by 

,_ W^ 

^ cos (a + ^S) (22) 

This reaction is, of course, the tooth pressure, and is expressed 
by the equation: 

,_ T^ 

^ tan (a; + ^) (23) 

cos (a + ^S) 
which may be written 




Fig. 8. 



•p 



np' = 



T 



[tan (a + ^)] [cos (a + ^)] 
T 



(24) 

(25) 



sm (a + 9) 

Messrs. Brown & Sharpe arrive at the same result in a different 
manner. Ignoring friction, they resolve the acting forces as 
shown in Fig. 8. 

T 



when 
hence 



2?' = 



cos X 

T _^ 
sin 90— X 



(26) 
(27) 



PRESSURE ANGLE AND FORM OF THREAD 29 

Neither of these equations, however, takes into account the 
pressure angle; for the more accurate expression see Equa- 
tion 43, Chapter VII. 

Many forms of worm teeth are possible. Hitherto we have 
considered only straight-sided teeth but the worm thread may 
be given any form, for example, cycloidal or involute, but the 
difficulty of manufacturing such worms as these is almost pro- 
hibitive and a straight tooth is the only alternative. A curved 
tooth worm was recently made and tested by the Brown & 
Sharpe Manufacturing Company in the experiments which 
they carried out early in 1912, but beyond being an extremely 
interesting mechanical production and working quite satis- 
factorily, the worm was of no special value. Much more impor- 
tant, however, is the hollow or Hindley worm, in which the worm 
being of hour-glass shape is made to embrace an arc of the worm 
wheel. That perfectly satisfactorily working gears may be 
designed on this system is proved by the uniform success ob- 
tained by such makers as Lanchester and Daimler. There is, 
however, a good deal of misconception as to the relative merits 
of straight and curved worms which it is the object of the 
following remarks to correct. As the author is in the position 
of having employed both types, he will, it is to be hoped, be 
exonerated from any prejudice in the matter. 

In the first place, when hobbing the wheel for an ordinary 
straight worm, the blank is entirely grooved out from a solid 
mass of metal by the hob in such a way that the hob completely 
generates the teeth of the wheel and as has been pointed out, 
every part of the face of each tooth is forced to touch against 
some part of the worm thread. The entire surface of the tooth 
is, therefore, subject to wear. The hob is, in this case, at its 
finishing end, an exact duplicate in its profile of the worm itself. 

If a hob be made of the curved form and a blank be cut with 
it, the tooth of the resulting wheel will be perfectly straight 
and the wheel will be flat on the points of the teeth. Fig. 9 
is a photograph of such a wheel cut by Brown & Sharpe for 
the experiments already referred to. It is seen that the worm 
is really the wheel, and the wheel, the ordinary straight worm, 
the relative diameters of the two having been reversed from the 



30 



WORM GEARS 



usual. Such a gear works well as shown by the experimental 
results. So far as the author's information goes, however, it is 
never used in automobiles. It is, nevertheless, the true Hind- 
ley worm, and, as a section through it shows, it makes contact 
at all points across the teeth, the entire tooth being subject to 
wear exactly as is the case with the straight gear, which in 
effect it is. 




Fig. 9. 

The gear first employed by Lanchester is not, however, a 
truly developed gear at all as both the worm and the wheel are 
hollow and such a wheel can only be hohhed with a very short hob 
and driven with an equally short worm. It has, however, this 
peculiarity that a section on the center of the worm shows a 
perfect contact along all the worm teeth at once. In other 
words, the worm applied to the wheel will shut out daylight 
entirely along its center line and if say four teeth of the wheel 
are subtended by the worm, all will be in contact at once. In 



PRESSURE ANGLE AND FORM OF THREAD 31 

Fig. 10, which represents any worm gear, let the lines A, B,C, D 
and E represent planes of section. Then in a Lanchester worm, 
a section of plane C will show every tooth in contact over its 
whole depth. Similar sections on B and D show decreasing 
contact, and sections on A and E show no contact at all. If 
the same section be taken on a straight worm gear, it will be 
found that on each plane there is one tooth in contact with the 
worm at some point across the tooth. Consequently with a 



ABODE 




Fig. 10. 

very little care on the part of the designer, the number of teeth 
in contact can be made exactly the same in the case of the 
hollow and the straight worm. This point must be insisted 
upon. Here, however, there comes a difference. The straight 
worm perpetually re-generates the surface of the wheel tooth 
maintaining the correct involute outline to the end of its life, 
and in the other type, as the pressure is always along the 
central plane, it follows that the greatest wear takes place along 
this line and the tendency is to finally reduce the worm wheel 
to the form of Fig. 11. The word 'tendency" is used advisedly 
because the rate of wear on any perfectly made worm gear is 
almost inappreciable so that the difference between the two is 
rather an academic one. One advantage is. on the side of the 
Lanchester worm; since the sides of the wheel never touch the 



32 



WORM GEARS 



worm, they can be cut away altogether and the wheel con- 
siderably reduced in width and weight thereby. Its disadvan- 
tage is that the worm cannot by any mechanical device be 
ground on any of its finished working surfaces, so less accurate 
finish must follow. 




Fig. 11. 

Another argument presents itself against the use of this type : 
it is that it is harder to assemble and adjust, since the worm 
must be truly located on the wheel in three planes, while the 
straight one requires adjustment in only two. This necessity 
for exact location in the axial direction of the worm demands 



PRESSURE ANGLE AND FORM OF THREAD 33 

much more care of the thrust bearings and further when the 
worm becomes warm under use expansion must take place 
endways and the contact be impaired. The fact remains, 
however, that this form of gear is entirely satisfactory if per- 
fectly made and in point of efficiency, the experiments of 
the Brown & Sharpe Manufactiu-ing Company^ referred to in 
Chapter X show that the efficiency of the hollow worm ap- 
proaches that of the straight so nearly that the difference may 
be said to be negligible and it is extremely doubtful if a suffi- 
cient number of examples were taken whether this difference 
would as an average exist at all. 

^ The Brown & Sharpe experiments referred to have recently (Nov., 1912) 
been confirmed by a very careful test on a Daimler worm driven axle, 
carried out by the National Physical Laboratory, London, of which a 
report has been published by the Daimler Motor Car Co. of Coventry, 
England. 



CHAPTER VI 
STRENGTH OF WORM-WHEEL TEETH 

The behavior of a worm wheel, considered as a structure, is 
exactly analogous to that of a spur wheel; that is to say, the 
stresses upon the teeth have precisely similar effects in their 
tendency to shear off the teeth, and therefore, apart from the 
special nature of worm gear and the effects of rubbing, which 
have to be provided against by special proportioning of the 
teeth, it becomes necessary to design them from this point of 
view in the first instance, afterward modifying the form of the 
teeth if necessary to suit the other considerations of the case. 




Fig. 12. 

The stresses in the worm wheel caused by the tangential 
pressure between it and the worm are limited, not by the horse- 
power of the motor but by the maximum tractive resistance 
between the car wheels and the road, thus 



V 



R 



(28) 



The tooth of the worm wheel is a cantilever on which the load 
w is applied at a point near the center of the tooth on the pitch 
line pi (Fig. 12). 

34 



STRENGTH OF WORM-WHEEL TEETH 35 

Let Z = length of thread of worm per . revolution through 

360 degrees. 
/? = angle of worm circumference subtended by worm 

wheel. 
>l = actual length of -tooth. 

P 

T =-- thickness of tooth on pitch line = — 

Then >l = ^ (29) 

The length of the cantilever k is the dedendum of the tooth 
which may be taken as .3683P. 

Let a = permissible stress in metal, say 7000 lb. per square 
inch. 

Then the safe load applied to the cantilever = 

W = — (where Z = modulus of the section) (30) 

K 

and ^^T ^^^^ 

If x = safe load on tooth in pounds 

'^"6^360^''^'' (32) 



X 




(33) 



•3683P 
This simplifies into 

a; = 2.2 Z^P (34) 

This formula supplies a very useful check on the design of the 
wheel and should always be applied before finally settling the 
dimensions of the gear. Except in very unusual cases, a wheel 
properly proportioned as regards the subtended angle /? and the 
number of teeth in contact, will be strong enough in any case as 
a spur wheel; the exceptions exist in very heavily loaded com- 
mercial vehicles where one of the brakes is applied to some part 
of the transmission system, whereby, should the wheels be 
locked by this brake, and the car be caused to skid, the tractive 



36 WORM GEARS 

resistance will rise to a high figure tending to shear off or at 
least to bend the wheel teeth, while at the same time, considered 
purely as a worm gear, the conditions are not by any means 
destructive for the high tooth pressure occurs with zero rubbing 
velocity and consequently there are no heating effects. 



CHAPTER VII 

STRESSES m WORM GEARING 

In a subsequent chapter, the stresses in the gears as affected 
by the width of the wheel and the number of teeth in contact 
will be investigated, but before considering these, it will be 
advisable to analyze all the stresses in the gears, as we can then 
determine the specific tooth pressure between the working 
surfaces and this value will be needed in the final proportioning 
of the wheel. 




Fig. 13. 

In the example already considered, the contact between the 
worm and the wheel is made at five different points simultane- 
ously, but owing to the fact that there are five threads to the 
worm, any diagrammatic representation of the same is much 
complicated; let us, therefore, consider the same worm divested 
of four of its threads as in Fig. 13; then some part of the remain- 
ing thread is brought in contact with every part of the surface 
of the wheel tooth against which it presses, and the successive 
lines of pressiu"e commence at a and end at /. The surface of 
the worm thread forms a continuous spiral around the axis, and, 
therefore, each one of these lines of pressure makes successively 
the same angle with the axis of the worm. In Fig. 13, these 

37 



38 



WORM GEARS 



lines of pressure are shown as a skeleton diagram, and it is seen 
that they form a portion of the surface of a conical spiral 
encircling the worm. Taking Fig. 13, the portion of this surface 
will be seen to be approximately a segment of a cone, and seeing 
that several teeth operate at one time, the central tooth in 
contact is touching the worm at a point which lies in a line 
joining the centers of the worm and worm wheel drawn at right 
angles to the former; the other teeth approaching to and receding 




Fig. 14. 



from this line, touch the worm threads respectively to the 
right and left of this point, viewed from the end of the worm 
axis. The lines of pressure, therefore, for every point of con- 
tact range themselves in such directions as will conform more 
or less to the surface of a cone, and the resultant line of pressure 
coincides with that of the tooth which is momentarily in the 
central position. Owing to the fact that this curved surface 
is not a true cone but a conical spiral, it will be appreciated that 
this is not mathematically accurate but sufficiently so for our 



STRESSES IN WORM GEARING 39 

purpose. Taking this central line of pressure the calculation 
of the direction of this force becomes exceedingly simple. 

Let ab, Fig. 14, be the axis of the worm and ah c d {t:^) Si per- 
pendicular plane cutting this axis; also c d e f {n^ a horizontal 
plane normal to n^ and touching both pitch lines; c d g his the 
cylinder of the pitch circle of the worm, and I jp m that of the 
wheel; p is the point of contact between the worm thread and 
the central tooth in engagement; p s is the line of pressure due 
to the angularity of the lead angle a^pris the normal pressure 

W 
angle elevated — degrees above the horizontal plane. Draw r o 

Zi 

and s 0, each perpendicular to the line d c; raise the perpendiculars 
r q and s q, and through the point q where they cut, draw the 
line p q; p q is then the resultant pressure line; p q does not lie in 
the plane of the axis a b and therefore p q and a h will never meet 
although not parallel to each other; it is evident that the force 
tending to separate the worm and the wheel passes along the 
line p q and is the reaction of the tooth pressure at this point. 

r s q is, by construction, a parallelogram; hence, p s qis a, 
right angle and if 2? 5 be drawn to such a scale that it represents 
the magnitude of the tooth pressure, 

p'= . / . N (25) Chapter V 

^ sm(a + ^) 

The length of the resultant p q can be calculated as follows 
Taking first, plane n^ (Fig. 14) 
p s = 90 degrees 
.'. p = ps cos a 
and so =ps sin a 



In plane tt^. 



or = po tan — (35) 

W 

and pr = po sec — (36) 

W 

. • . or = ps cos a tan — (37) 

¥ 

and pr = ps cos a sec — (38) 

<6 



40 WORM GEARS 

The triangle pqs can then be resolved as follows: 
qs = or and psq = 90 degrees 
.\ {pqy={ory-\-{psy (39) 

Substituting (pqy=(ps cos a tan - 1 + (psy (40) 

.'. pq = ^^{ps cos atsiii -^ j +(ps)' (41) 

T 

and since ps is by construction = p' = 



sin (a + 9?) 

we at once obtain the value of pq the resultant pressure, and 
since pq is greater than ps, the value of pq must be taken 
wherever p' is in question^ in terms of the torque T. 

, (T cos a ^ r\2 /TV 

pg=Aj(7'cotatan|y+(^-j^)V (43) 

and pq must be taken as the correct value of p' wherever the 
latter is required to express the actual tooth pressure normal to 
the working surface. 



Lastly, qs = ps tan z 



and substituting 



tan 2 = -- (44) 

ps 



tan 2 = 



p cos a tan-^ 



(45) 



P' 



¥ 

tan ^-=cos a tan -^ (46) 

Whence z is easily found from Equation 20, Chapter V. 

This settles the direction and magnitude of the resultant 
line of pressure due to the torque of the motor. 

The worm and worm wheel are carried in ball bearings in 
which the stresses can be found in the following manner. The 
end thrust of the worm in the direction of its axis is po (Fig. 16) 
and is equal to p' cos a. 

1 See Equations 25, 26 and 27.. . , 





STRESSES IN WORM GEARING 


but 


T 


^ sin {oL-\-(p) 


Therefore, 


T cos a 

V0= • / , N 



41 



(47) 




Fig. 15, 



It has already been shown that cp is negHgibly small. 

T cos a 



Therefore, 



po 



sm a 



Tcota 



(48) 



po is the tangential pressure on the worm = p 

.\p=T cot a (49) 

Let /=pg = the separating force, qs = ro = ihe force acting 
radially on the journal bearings of the worm. 





/ 


S 


7 


\ 




^1 






r 



Fig. 16. 



or = p^ COS a tan 



¥ 



and substituting for p^ 

or= 



T , W 

— — -, — , — r- cos a tan -^ 
sm {oL + (p) 2 



(50) 



(51) 



42 



WORM GEARS 



neglecting <p 



T cos a tan 



W 



or = 



sin a 



.'. or=T cofc a tan 



¥ 



(52) 
(53) 



. Notwithstanding that this vertical force is the actual pres- 
sure tending to separate the worm and wheel in the direction of 
a line drawn between and perpendicular to each of their axes, 
it is not, as Mr. Kalb has shown, the ultimate stress on the 
journal bearings for the following reason. In Fig. 17, let A be 




Fig. 17. 

the axis of the worm, T is the moment of torque at the pitch 
line; it is, of course, balanced by the reaction R which is the 
horizontal stress upon the journal bearings of the worm =7". 
We have just seen, however, that there is a vertical pressure on 

¥ 

these same bearings = T cot atan— = Fin the figiu*e. There 

is, therefore, a total stress on the journal bearings which is 
equal to the resultant of F and R, viz., S; from the construction 
the figure AB is a rectangle, hence 



substituting for F^ 



S'^F'^-R'' 



¥' 



S'={T cot a tan-) H T^ 



S = 



=v 



(Tcotat&n^y + T' 



(54) 

(55) 
(56) 



STRESSES IN WORM GEARING 



43 



S being in this case the resultant separating force. It is evident 
that this force acting on the bearings of the worm has its exact 
counterpart in the bearings of the worm wheel; in Fig. 18, the 
two side components of the forces acting on the wheel are 
shown, and they are seen to consist of the torque T and the 
vertical pressure F as before; the resultant force S produces a 
diagonal stress on the worm wheel and differential case, but as 
the vertical and horizontal components are located in the plane 




Fig. 18. 

of the axis of motion in the case of the wheel, there is no result- 
ant force acting on the bearings. It follows from this, therefore, 
that theoretically the journal bearings of the wheel are more 
lightly stressed than those of the worm, but in practice it is not 
possible to make them any smaller, even if it were desirable, 
since the arms of the differential case are invariably larger in 
diameter than the worm spindle and necessitate a larger bearing 
to support them. 



44 WORM GEARS 

With regard to the side thrust of the worm wheel, referring 
to Fig. 14, we see that the component of side thrust of the force 
pq is in the direction so, we know this to be = ps sin a,=p^ 
sin a 

T 

but p'= -. — - — -^ 

^1 » T sin a 

tJiereiore so= —. — ; r- 

sm {a + cp) 

neglecting <p as before 

so = T 

It has already been pointed out that the stresses in the gears 
due to braking, especially in the case of a heavy commercial 
vehicle, may be very much greater than those due to the motor 
torque. Just before the wheels come to rest by skidding 
with the application of the transmission brake, the end thrust 
on the worm must be considered to be that due to the tractive 
resistance at the road wheels, multiplied by the radius of the 
road wheels, and divided by the radius of the worm wheel. As 
however, at the instant of coming to rest, that is when the 
tractive effort is so low as to be negligible, the motion of the 
worm is very slow, the ball bearings of the worm thrust, being 
scarcely moving, will carry a very great load, and do not there- 
fore need to be very much greater than is necessary for satis- 
factory working with more moderate loads at high speeds. The 
conditions under which the vehicle will be required to work are, 
however, so various that the risks which may be taken in this 
respect must be a matter of the designer's judgment and 
experience, and no empirical rules can be laid down for them. 

It need hardly be pointed out that the torque T is the torque 
delivered at the pitch line of the worm when the lowest trans- 
mission gear is in use, and though the value of p' v does not 
become any greater (but rather less, owing to the work absorbed 
in tooth and bearing friction in the transmission gears) p' is 
greater than the torque of the motor itself, by the amount of the 
gear ratio in operation. 

We are now in a position to tabulate the complete stresses 
in worm gearing for any angle of worm gear, and their magni- 
tude per pound of torque moment for all usual angles of lead is 



STRESSES IN WORM GEARING 



45 



given in the following table, where the axial pressure angle is 
assumed to be 30 degrees. In this table the value of the fric- 



Curves showing Stresses per Pound of Torque 
for Various Lead Angles 
Axial i'ressure Angle = 30 ° 




16 IS 20 22 24 26 28 30 32 34 36 38 40 42 44 46 
Degrees Lead Angle a 

Fig. 19. 

tion angle has been omitted as being too small to have any 
practical value. 



46 



WORM GEARS 



Fig. 19 has been plotted from this table, and from it all the 
intermediate values of different angles of lead may be found. 
This diagram will be found of great value in designing or 
analyzing the stresses in any worm gear. 

TABLE III.— STRESSES IN WORM GEARING 



igle a 




Worm 


¥ 


end 
thrust 


15° 


58° 18' 


3.732 


20° 


56° 50' 


2.747 


25° 


55° 14' 


2.144 


30° 


53° 8' 


1.732 


35° 


50° 38' 


1.428 


40° 


47° 44' 


1.191 


45° 


44° 26' 


1.000 



Resultant 




Result- 


normal 


Separat- 


ant 


tooth 


ing force 


radial 


pressure 




thrust 



Side 
thrust 



4.39 
3.27 
2.61 
2.18 
1.88 
1.64 
1.47 



2.16 


2.38 




1.485 


1.79 




1.122 


1.52 




0.866 


1.322 




0.676 


1.202 




0.527 


1.130 




0.405 


1.080 





Lead angle = a 

w e 

¥ is obtained from tan ^ = tan ^ cos a 
Worm end thrust = T cot a 



(20) Chapter V 
(49) 



Normal tooth pressure = yJ{T cot a tan ^ ) + 
Separating force = T cot a tan 



T \2 



sm a, 



¥ 



Radial thrust 

See Appendix A. 



¥\^ 



= \I(T cot aiding] +r 



(43) 
(53) 
(56) 



CHAPTER VIII 

THE WIDTH OF THE WORM WHEEL 

When an automobile is being driven by the motor, the torque 
of the latter is applied to the road wheels up to the point where 
they will commence to slip; this slip will prevent the increase 
of horse-power transmitted beyond a certain fixed amount (see 
equation 28, Chapter VI). In the case of a very heavy vehicle, 
ihe tractive effort is much greater than the motor torque and 
the strength of the wheel teeth x (equation 34, Chapter VI) 
must be such that it will be sufficient to withstand the tooth 
pressure due to stresses of braking calculated from equation 28. 

There are always more than one tooth in contact at one time. 
Consequently, 

x = --, and we can write (59) 

n f ^ 

2.2 Z/?P = -^7 
n 

A o V' (60) 

^^^^=2TyPn' 

^ "" 2.2 Z P 

It is evident from this that /? and n' vary inversely, and /? 
cannot be diminished without correspondingly increasing n', 
and since n' from the design of the gear is unalterable, it follows 
that if /? be reduced, the equation can no longer be satisfied. 

The alteration of /9, therefore, has a very important effect on 
the behavior of the gears, since the specific pressure would by 
its decrease be augmented. B is thus given a new and impor- 
tant significance, for in addition to determining the strength of 
each individual tooth, it also governs the number of teeth in 
engagement at one time. 

47 



48 



WORM GEARS 



Mr. John Younger has suppHed the following means of obtain- 
ing the value of ^. 

The length of the worm g is a, multiple n^ of the circular 
pitch P, and it contains in its length a certain number of 
threads n^ of which it is most important to take the fullest pos- 
sible advantage. In order to ensure this, it is necessary to de- 
termine the width of worm surface, or in other words, the angle 
^, which will provide a wheel tooth to each thread of the worm. 
This can be determined graphically in the following manner: 




Fig. 20. 



Let A B in Fig. 20 be the length of the worm = g; AC is the 
outside circumference of the worm. From the point A, AD 
is drawn so that the angle DAC =a. From B, the line BE 
is drawn perpendicular to AD. The distance A E is the length 
of the ^,rc of the worm which must be subtended by the worm 
wheel in order that some part of every thread of the worm in 
the length AB will touch one of the wheel teeth; from this 
diagram, the following equation is obtained from which the 
value of /? is at once found for any example. 



/?= 



360 g tan a 

7i{d + 2xo) 



(62) 



2'HE WIDTH OF THE WORM WHEEL 



49 



when g is the total length of the worm = n'P 

2XO-2X.3183 P-.6366 P (63) 

Hence the equation becomes 

360 g tan a (64) 

^ 7r(d + .6366P) 

In Fig. 21 the cross-section of the worm wheel is shown. It 
will be observed that the '^ width" of the wheel can be measured 
at three points, viz., at the pitch line, at the root of the teeth, 
and at the outside, this last being somew^hat deceptive as it has 
no natural connection with the width of the working face. 




Fig. 21. 



Each dimension may be expressed as the chord of the angle 
/? measured at different radii. The radius of the pitch line 

being 



d d 

- that of the root of the teeth -^-{-k. 



Sufficient thickness of metal must be left in the rim to prop- 
erly support the teeth of the wheel as in the case of a spur gear. 

The subject of worm contact has been very fully investigated 
by Mr. Robert A. Bruce (Proc. Inst. Mech. E., Jan., 1906), and 
the reader is referred to this for a very complete geometrical 
analysis of the subject; the present writer, however, takes 
exception to some of his reasoning which is somewhat confusing 
by his method of illustrating the contact by the use of vertical 
parallel planes cut through the worm teeth. Such planes do 
not follow the lines of pressure between the contact surfaces and 
are consequently distorted sections through what is admittedly 
a very difficult section to correctly illustrate upon paper. 



50 WORM GEARS 

He points out, however, that every part of the outline of the 
wheel tooth is compelled at some time to touch some part of the 
worm. This is, of course, self-evident when it is remembered 
that the wheel is cut with a hob whose outline is- the exact 
replica of the worm. 

Mr. Bruce sums up his conclusions as follows : 

''The area of physical contact varies as the pitch diameter of 
the worm multiplied by the square root of the diameter of the 
wheel; or, if the effects of varying the angle subtended by the 
pitch line of the worm wheel at the center of the worm be con- 
sidered, it may be said that the effective area of contact varies 
as the continued product of the diameter of the worm, the 
tangent of half the angle subtended by the worm wheel, and the 
square root of the diameter of the worm wheel. At any instant 
the end pressure is shared between several teeth, and it is 
therefore justifiable to expect a greater power of sustaining 
loads as the number of teeth in action is greater. The variation 
in the number of teeth in gear is, however, much more apparent 
than real. Except in the case of abnormally small worm wheels, 
the length of the contact paths on the worm wheel side of the 
pitch plane is unaffected by the size of the worm wheel On 
the other side, the contact lines most affected are those which 
are flattest, or which most nearly coincide with the pitch line. 
The actual variation in the number of teeth in gear at any one 
time is found on careful investigation to be small for widely 
differing sizes of worm wheel. So that in comparison with 
other more important matters it may be neglected. 

" By keeping the ratio of the height to the thickness of the 
teeth as large as practicable, the greatest possible number of 
teeth are enabled to operate simultaneously, but at the same 
time, in order to avoid interference, the teeth should be pitched 
as finely as is compatible with strength and allowance for wear. 
It should be noticed that in respect of the height of the teeth 
the dictum given above is in direct opposition to the best prac- 
tice with spur gearing, where entirely different conditions are 
in force. 

''The effect of the angle of the worm thread remains for con- 
sideration. As the ratio of pitch to the diameter of worm be- 



THE WIDTH OF THE WORM WHEEL 51 

comes greater, the thrust of the worm is borne on a surface of 
greater incHnation and the actual pressure on the teeth is 
increased in the same ratio as the secant of the angular pitch. 
At the same time the width of the contact line across the face 
of the teeth is increased in the same ratio, so that the actual 
pressure per unit of width remains the same. It is not necessary 
therefore to take any account of tlie angle of tJie helix in makiiig 
estimates of the ^effective contact area'. Under precisely similar 
conditions as to temperature, lubrication, nature of the surfaces 
in contact and rubbing velocities, it might reasonably be 
anticipated that the end thrust would be proportional to the 
effective area, and neglecting comparatively unimportant 
factors the relation may be expressed as follows: 




V = K^^Ddt2iii^ (65) 

so 29 = some factor depending on conditions X effective breadth 

X effective width across the face of the worm teeth, 
where p =safe end- pressure in pounds. 

d = diameter of worm at pitch line. 

D = diameter of worm wheel pitch line. 

/? = angle subtended by the worm wheel at the axis of 
the worm. 

K =Si factor depending upon rubbing velocity, nature of 
the surfaces, temperature and nature of the lubri- 
cant, etc. 

''Experience has shown that this relationship is far from 
simple in practice, because of the great difference in the factor 
K imposed by variable conditions. Broadly speaking, what 
happens in the case of a worm wheel drive is very much what 
happens in the case of a loaded journal. The action commences 
under certain conditions as to speed, temperature and so forth, 
and as it proceeds heat is generated owing to frictional resistance, 
the amount depending upon the load, the lubricant and the 
efficiency of the gear. The temperature of the system rises 
until the heat generated by friction balances the heat lost by 
radiation and convection, when a stable set of conditions is 



52 WORM GEARS 

established. But while the temperature is rising, the lubricant 
is losing its viscosity, and, though this tends to diminish the 
friction and consequently the generation of heat, it nevertheless 
diminishes the power of sustaining a heavy load. A worm will, 
therefore, be successful if the viscosity of the lubricant does not 
diminish to such an extent that its load-sustaining properties 
are neutralized. If the surfaces be allowed to come into grind- 
ing contact, further heating takes place, and the lubricant 
becomes still less viscous and therefore incapable of bearing a 
load, and seizing will take place quickly." 

Mr. Bruce, in his paper already referred to, gives the follow- 
ing formula for calculating the width of contact area from 
which he determines the tooth pressure. According to this, 



6 = ZA/-^^^i (66) 

when fj and r^ are the respective radii of the surfaces in contact. 
As, however, owing to exigencies of interference when being 
hobbed, it is practically impossible to determine the radius of 
the wheel-teeth surfaces, this formula does not seem to have 
much more than a theoretical value. Moreover, owing to the 
elasticity of the metals employed, and variations in the quality 
of the lubricant, no reasonable figure can be assigned to K in 
the equation. All that we can do, therefore, is to establish 
from experience an empirical value for the number of teeth in 
contact in its relation to actual performance, and deduce some 
rule from this. 

The author carried out the following experiment to ascertain 
the effect of varying /? in a specific instance. 

An axle was taken fitted with a standard worm gear similar 
to many hundreds running in heavy commercial vehicles of the 
author's design and giving eminently satisfactory results. The 
gear was driven from a 50-H. P. electric motor running at a 
constant speed, and an hydraulic dynamometer was attached 
to the other side of the gear, and adjusted to furnish a constant 
load (Fig. 21 A). The dimensions of the worm gear were as 
follows : 



THE WIDTH OF THE WORM WHEEL 



53 




< 
6 



54 



WORM GEARS 



Worm : Number of threads 5 

Pitch diameter 3.25 in. 

Pitch circumference 10.21 in. 

Length of worm 5 in.' 

Lead 5. 9375 in. 

Lead angle 30° 11' 

Thread angle 60° 

Length of thread per rev 11.812 in. 

Rubbing speed 18.5/. p. s. 

Worm Wheel: No. of teeth 39 

Pitch diameter 14.75 in. 

Pitch 1.875 in. 

Subtended angle /? (see table). 

The torque at the worm pitch Kne was kept constant at 
1990 lb. giving a normal tooth pressure of 4378 lb. The rubbing 




A EI 



A EI 



LHD 




BFJ 



Fig. 22. 



speed was also nearly constant at 18.5 ft. per second. Conse- 
quently the heat generated in friction (from equation 68^ Chap- 
ter IX) was 12.6 B.T.U. per minute and (from equation 73, 
Chapter IX) the area of rear axle surface should be 8.6 sq. ft. 



THE WIDTH OF THE WORM WHEEL 



55 



The actual surface was approximately 11 sq. ft. Fig. 22 shows 
a section of the worm wheel of which the original value of /? 
was 107° 45'; under these conditions, the number of teeth in 
contact was 5 (equation 12, Chapter IV). The wheel was subse- 
quently cut down in width as shown in the figure. 

In Fig. 22, the gears are shown diametrically in section and 
plan, the lines AB, CD, indicate the original width of the gears 
with /? = 107° 45'. There are then five points of contact — a, h, 
c, d, and e. 

/? was then reduced to 72° as shown by lines EF and GH, 
The points of contact are thus reduced to three — h, c, and d. 

/? reduced again to 64°, still left three points of contact, but 
further reduction to 55° 30' as shown by lines IJ, KL left only 
two teeth in contact at c. 

The following table shows the results obtained: 

TABLE IV 



No. 4 




Subtended angle /? 

Revo, of worm 

Rubbing velocity v. f .p.s 

Tooth pressure p' 

Area at base of tooth 

Width of wheel 

Temperature attained t' 

Temperature of air t 

t'-t... 

No. of teeth in contact 

Pressure per tooth 

Estimated pressure per square 
inch. 



107° 45' 


72° 


64° 


2,000,211 


775,008 


1,590,000 


18.5 


18.25 


18.34 


4378 


4378 


4378 


3.286 in. 


2 . 262 in. 


2 . 1 in. 


3.25 in. 


2 . 3437 in. 


2.0937 in. 


163° F. 


412° F. 


412° F. 


75° 


76° 


74° 


88° 


336° 


338° 


5 


3 


3 


875 


1459 


1459 


5840 


9730 


9730 



55° 30' 
Failed 

18.34 

4378 

1.82 in. 

1 . 8437 in. 

500° F. 

72° 

428° 
2 

2189 

14,580 



The failirre of the gears in Experiment No. 4 took place from 
partial seizure. In the first three experiments, the duration 
was sufficient to arrive at constant conditions and a maximum 
temperature rise, but about twenty minutes running under 
conditions No. 4 showed the temperature rising so rapidly that 
operations were suspended and the gears examined. It was 
plainly evident that failure was due to actual metallic contact 



56 WORM GEARS 

between the surfaces, and this fact taken in conjunction with 
the last hnes of the table shows an unmistakable connection 
between specific tooth pressure and temperature. The condi- 
tions in No. 1 test being alone normal when the individual tooth 
pressiu-e was 875 lb. 

To determine the area in contact on each point is impossible 
with any accuracy but in the author's opinion, in the gears in 
question, this amounted to .15 to .2 sq. in — probably the 
former. Assuming this figure, we should have in the first 
experiment 5840 lb. per square inch, and since experiment shows 




Fig. 23. 

that the dry metal will carry this pressure within its elastic 
limit, there is every reason to believe it was not much exceeded. 
Plotting the pounds per tooth in the form of a curve (Fig. 24), 
we can obtain an approximation to the intermediate points, 
and we find the curve to be distinctly hyperbolic in character as 
might be expected, having its o origin with an infinite number 
of teeth and rising to infinity with o teeth. 

All that can be said positively about the contact is that 
undoubtedly the surfaces are separated by a film of oil (Fig. 23), 
and between the two convex surfaces the existence of this oil 
cushion can be relied upon provided an individual tooth pressure 
of say 900 lb. is not exceeded at a velocity of 18.5 ft. per second. 
Multiplying these values together, we at once obtain the neces- 
sary constant, 

p''y = 16,650 (67) 

and it may be safely said that 17,000 must never be exceeded for 
continuous running if overheating is to be avoided. 

The fact will not be lost sight of that in all four of the above 
mentioned experiments, the quantity of heat generated was 



THE WIDTH OF THE WORM WHEEL 



57 



approximately constant, and of course, the surface of radiation 
remained the same; the convection and radiation of heat must, 
therefore, have been at a constant rate also, and the rise in 
temperature was due to an alteration in the coefficient of fric- 



2800 



2600 



4^2400 

I 

^3 2200 



i3 
C2000 

<o 

6 
o 

?1800 



o 1600 



1400 



u 1200 
m 
^ 1000 



800 



600 



12 3 4 5 

No. of Teeth in Contact 

Fig. 24. 

tion, caused by the, at first, partial, and subsequent total, 
breaking down of the oil film with the increased tooth pressure. 
Too much stress cannot, therefore, be laid upon the importance 
of securing a proper lubricant. In these trials, the oil used was 
what is commercially known as ''600 W cylinder oil," this 
being a mixture of heavy mineral oil of high viscosity with 
some animal oil and possessing the following properties: 

Beaum6 gravity 25 . 9° at 60° F. 
Specific gravity .8980 at 60° F. 

Viscosity 154.4 at 210° F. 

The viscosity reading is with a Tagliabue viscosimeter. 
It was found that a sample of this oil could be heated to 
700° F. in an open cup without igniting. 



CHAPTER IX 

THE TEMPERATURE COEFFICIENT 

In any worm gear, there must always be a definite amount of 
heat generated since it cannot operate without some friction, 
which is converted into heat. 

Let H be the quantity of heat in B.T.U. per minute. 

Then ^ = ^? (68) 

This quantity of heat is added every minute to the complete 
axle system which will raise its temperature to a definite extent; 
and if /i = mean specific heat of the whole axle system and m 
its mass in pounds, its capacity for absorbing this heat will be 
mh. Then if no radiation takes place from the axle, its tem- 
perature will increase at the rate of 

TJ 

— T degrees F. per minute (69) 

It is evident that the heat generated has ultimately to be 
radiated from the axle at the same rate per minute. Let g = 
this quantity of heat in B.T.U. 

It is desirable that the oil should not be permitted to attain a 
temperature of over 180° F. because its viscosity and lubricating 
properties are considerably reduced at higher temperatures, 
and if we assume the worst conditions and provide for an atmos- 
pheric temperature of 100° F., it follows that the difference 
between the temperature of the worm and the atmosphere will 
be 80° F. 

Let f and t respectively be the temperatures of the gear and 
the atmosphere. Then 

/'-^ = 80° 

58 



THE TEMPERATURE COEFFICIENT 59 

A formula for radiation is 

q= (t- ti) XaXc (70) 

Where a = radiating area in (J and c is a figure derived 
from experiment denoting the B.T.U. radiated per minute 
per square foot per degree F. difference. This may be 
taken as 

.455 at 15 f.p.s. velocity of cooling air. 
Whence q = 36.5a, B.T.U. per minute (71) 

Connecting equations (68) and (71) 

^i^ = 36.5a 

or a = .t)000423 p'v (72) 

if p'v = 15,000, then a = .635 sq. ft. (73) 

the minimum exposed area of the axle. 

This figure is necessarily only approximate because the rate 
of radiation varies with so many factors. For example, the 
condition of cleanliness or otherwise of the axle casing and its 
speed through the air, atmospheric conditions, and so forth. 
Moreover, heat is conducted away from the gearing through the 
axles and springs and to some extent through the propeller 
shaft. It will be found in practice that the surface of radiation 
determined by the formula is quite easy to obtain, but if from 
any cause it is too small, overheating will be only a question of 
how often the vehicle has to run at maximum power on low 
gear, when conditions of tooth pressure will be highest. Even 
in this direction, some chances may legitimately be taken, since 
continuous running on low gear very rarely occurs, and even 
on long hills there are periods where the maximum horse-power 
is not exerted, which will consequently tend to a heat balance 
being maintained. 

It must be pointed out that the oil used for lubrication is but 
a poor conductor of heat so that the difference in temperature 
between the gears and the atmosphere is generally greater than 
is rendered apparent by a thermometer placed in the oil, but 



60 



WORM GEARS 



the gears may be run for short periods at very much higher 
temperatures than those allowed for. 

In formula (68) since /i is a constant quantity, viz., .002, it 
follows that 



H = Kp'V 
when K is a constant = .000154. 



(74) 



3500 



3000 



Si, 



0^2500 



(k 



2000 



1500 



1000 



















































































V 






































































\ 








































































V 






































































\ 








































































V 






































































\ 








































































\ 








































































s/ 






































































S 


v 






































































\ 








































































\ 


p'l 


;= 


5C 


>600 




























































\ 








































































\ 










t 






























































\ 








































































\ 


s 








































































\ 








































































\ 


V 








































































\ 


X 








































































■V, 


V 


















































































































































*^ 


--«. 


-«, 










































































— 












































































^ 


■-. 









































































10 15 20 25 30 35 

Velocities - Ft. per Sec. 

Fig. 25. 



40 



45 



and consequently H varies as p^V, and if a constant value is 
demanded for H, it follows that p'F is a constant. 

In Fig. 25, an hypothetical case, the rubbing velocities are 
represented as abscissse and the tooth pressure in pounds as 
ordinates, p' 7 = 50, 600 a constant and the curve is in conse- 
quence an hyperbola. 

In the Bach & Roser experiments, it was found that at 
constant temperatures, a very close approximation to this 
hyperbola was obtained. Thus, the important deduction from 
this is, that given the maximum pressure and speed of any gear 
at which a constant temperature may be maintained, the other 
pressures and speeds can always be calculated, lower speeds 
permitting higher pressures and conversely. 



THE TEMPERATURE COEFFICIENT 



61 



It is profitable to investigate the effect on the quantity of 
heat generated by varying the pressure and the velocity. 

pV 60 



H=v 



778 



Case 1. — Assume p' constant at 3500 lb. then 

^^ 3500X.002X60 ,, 
H = v ^^3 ^..54. 

TABLE V 



V f.p.s. 


B.T.U. per minute 


5 


2.70 


10 


5.4 


15 


8.1 


20 


10.8 


25 


13.5 


30 


16.2 


35 


18.9 


40 


21.6 


45 


24.3 



Here the difference = 2.7 B.T.U. per increment of 5 f.p.s. 
Case 2. — Assume v constant at 45 ft. per second 



H^2^' 



.002X45X60 

778 



.00694 / 



TABLE VI 



V' 


B.T.U. per minute 


1500 


10.41 


1750 


12.14 


2000 


13.88 


2250 


15.62 


2500 


17.35 


2750 


19.09 


3000 


20.81 


3250 


22.55 


3500 


24.30 


3750 


26.05 


4000 


27.75 



62 



WORM GEARS 



Here the difference = 7.4 B.T.U. per increment of 250 lb. Fig. 
26 shows these increments of H plotted against pressures and 
velocities respectively, and the results show that for a com- 




2500 § 
2250 ^ 
2000 g 
1750 o 
1500 ^ 



10 12 14 16 18 
B.T.U. per Minute 

Fig. 23. 



paratively small increase in the normal tooth pressure, the 
, heating effect is increased at a greater rate than with relatively 
larger increments of the velocity. 



CHAPTER X 

EFFICIENCY OF WORM GEARING 

The efficiency of worm gearing has been discussed by several 
writers and it will be dealt with next, because closely bound up 
with it is the coefficient of friction between the worm and 
wheel, and the determination of the latter is a necessary 
preliminary. 

The most complete investigation of mechanical friction that 
the author is familiar with, is that conducted by Beauchamp 
Tower, and published by him in various numbers of the Proceed- 
ings of the Institution of Mechanical Engineers. Taking the 
materials most nearly corresponding to those employed for 
worm gears, viz., steel rubbing against bronze in a bath of 
mineral oil, we find the lowest value he obtained for /i was .0008. 
No doubt these conditions were ideal, but suppose we take a 
higher value, .002, as an assumption, and then see how nearly 
this can be approached and maintained in practice; it will 
presently be shown that we are fully justified in so doing, see- 
ing that the conditions favorable to a low coefficient are all 
present in a worm gear. These are 

1. Brief period of contact. 

2. Intermittent pressure. 

3. Certainty of oil films reaching every part of the pressure 
surface. 

4. No metallic contact. 

The well-known experiments of Bach & Roser at the Royal 
Technical High School, Stuttgart, throw some light on the 
behavior of worm gears, but unfortunately they were carried out 
upon gears which were by no means as accurately made as is 
the case in the present day practice; but while their results are 
of little specific value, they undoubtedly illustrate the laws 
which govern the whole performance of worm gears. 

63 



64 WORM GEARS 

The worm gear upon which the experiments were made 
•appears to have been of the following proportions: 

Worm: Number of threads 3 

Pitch diameter 3 in. 

Lead 3 in. 

Lead angle 17° 40' 

Wheel: Number of teeth 30 

Pitch diameter 9 . 54 in 

Circumferential pitch 1 in. 

The tooth pressures were varied from 190 to 2660 lb., and tests 
were made at six different speeds as follows: 

V =28.30 f.p.s. 
19.34 f.p.s. 
9.80 f.p.s. 
4.62 f.p.s. 
2. 56 f.p.s. 
0.85 f.p.s. 

It will be observed that while the tooth pressures are m some 
accordance with automobile practice, the velocities are lower 
than is ordinarily the case. 

Fig. 27 shows the tooth pressures and coefficients of friction 
for different velocities obtained in these experiments. It is at 
once apparent that for velocities between 2.5 and 20 ft. per 
second, the coefficient is but little affected by the speed but 
varies curiously with the pressures on the teeth, attaining in 
all cases a minimum at or near 1000 lb. tangential pressure. 
Unfortunately these experiments do not state, or give sufficient 
data to calculate, the specific tooth pressure per square inch, 
which would have been very valuable. As has, however, been 
pointed out by Mr. Robert A. Bruce, the deductions from these 
experiments are only useful when the exact conditions of the 
original experiments are reproduced, and he further points out 
that with superior working surfaces, e.g., hardened and ground 
worms, and efficient cooling arrangements higher pressures 
might be realized. 

With a view to proving the comparability of worm gears 
and the journals investigated by Beauchamp Tower, the author 
carried out the following experiment. 

A completely assembled rear axle belonging to a 5-ton com- 



EFFICIENCY OF WORM GEARING 



65 



mercial vehicle of the author's design was taken and mounted 
upon a stand (Fig. 28), using one of the road wheels as a drum 
round which a weighted cord could be wound. A smaller 
drum was secured to the worm spindle with another weighted 
cord wound upon this. The relative diameters of the drum on 
the axle and the drum on the worm spindle were such that their 
ratio was the same as that of the worm gear reduction, viz., 7.8 
to 1. Equal weights being hung upon the cords, equilibrium 



0.07 



0.06- 



g0.05 
o 



m0.04 



6 0.03 



0.02 



0.01 



i\t;, 


















































ar 


























































w 
























































— 




\\V 


^V = 19.34 i.p 


.s. 












































w 




































c 




^ 












\ 


Y ^=28.3 


































J 


/ 














V 








































/ 


e 














\ 
























!^ 


to.85^ 








x^/ 




















\\\ 


















f^ 


y 












y 


-^ 


y 


' 
















- 




\ i\\ 




















\, 









-- 




/ 


V 
























V\ 


\ 




























,/ 


/ 


/ 
























Tv^\ 




V' 


= 9.8 


















/ 


/ 


/ 






























\ 






















/ 


/ 


/ 




























Tx 


> 


\^ 


V= 


= 4. 


62 






t 


I 


/ 


/ 


/ 


/ 


































vS^ 


S 


-6- 




>// 




A 


/ 






































^>^ 


- J^' 




X 










































kxVJ 


b^^ 




y 


y 










































/ ^ 


r^ 






.-- 


^ 












































4 


= 2.5^ 








































































BACH & ROSER 
Relation between Pressure, Velocity 
and Coefficient of Friction 




— 







































































































































































































































500 1000 1500 2000 

Tangential Tooth Pressures - Lbs. 
Fig. 27. 



2500 



3000 



was established. The mechanism being set in motion by hand, 
weights were added to the cord on the worm drum until a con- 
stant speed of rotation could be maintained. 

On substituting heavier weights, it was found that a greater 
weight was needed to keep the wheels in motion, and, several 
observations being thus made, a number of values w^ere obtained 
for the weight necessary to maintain uniform motion with 
different degrees of tooth pressure. Plotting these values at 
ordinates, upon tooth pressures as abscissae, it was found that 



66 



WORM GEARS 




00 



^ 



EFFICIENCY OF WORM GEARING 



67 



these fell in a straight line as shown by the line A (Fig. 29) 
The observed points being 



TABLE VII 



Pressures x 


Load to overcome friction y 


2 1b. 
4 1b. 
8 1b. 


3.75 
4.75 
6.75 



It is evident that the loads y are the product of a;X/^ 





Curves showing Values of jJ- from Observed Friction Readings 
from Observed Points. V = Xti &. 2y-X=5.5 .'. 2XfJi.-x=5.5 
fi approaches V2 as a Limit as X approaches co 
H .. CO .. .. « ., .. .. Zero. 


7 


























































/ 


' 










































6 

3. 


3 






/ 












































■?. 


n 


/ 


f 












































h 


01 


-) 


/ 


A 















































V 


f 














































•m 4 
o 




/ 
















































y 


/ 
















































s 


/ 


















































1 


















































^z 


\ 






















2X 
























\ 


I 








































1 




\ 




















































"^ 


■ — . 













































n 





















































2 4 6 8 10 



20 30 

Pressure in #iC 

Fig. 29. 



40 



50 



The equation to a straight line is 

2y — x = 5.5 
.*. 2xfi — x = 5.5 

5.5 + x 

Fig. 29 shows the curve to this equation giving values of /x 
for various pressures x, from which we see that 



68 



WORM GEARS 



[i approaches . 5 as a limit as x approaches c-o 
H approaches oo as a Hmit as x approaches o. 

Turning again to Beauchamp Tower's experiments, the 
curves in Fig. 30 show the relationship between bearing pres- 
siu-es and n for three different rubbing velocities, from which it 
is evident that the characteristic of the curve obtained from the 
entire axle is identical with these and consequently the rules 
deduced from Tower's experiments by Unwin for the variations 
of ji with pressures and speeds will apply equally to worm 
gears. This is highly important in all that follows. 

Beauchamp Tower's observed figures for mineral oil bath lubrication. 

Frictional resistance 2? = /fP = P(7-\ /— 

Mean value of C for mineral oil .27 
0.012 

O.QU 

0.010 

ci 0.009 

% 0.008 

2 0.007 

M 0.006 
•♦J 

.2 0.005 
g 0.004 

u 

0.003 



0.002 
0.001 
0.000 















(a) 0.426 0.26 
(b^ 0.56 0.27 
(C) 0.733 0.28 
















"\ 


^ 






h 


X<? 


(c) a approaches 0.0023 as Limit 
(b) " " 0.0017 " " 
(a) " " 0.0014 " •« 
As X approaches co 
<( << i( •< 
<( « <« <« 


\ 




^\^ 








-^^ 








c^^~~^i^ir — 


— 






^ — ^ 













100 200 

Bearing Pressures, Lbs. a " 

X 

Fig. 30. 



300 



400 



It will at once be objected that the values of // obtained by 
experiment are much greater than .002; the reason for this is 
very plain. In the axle, as tested, there was the friction of the 
ball bearings supporting the worm and the worm wheel, and 
also the friction of the roller bearings on which the heavy road 
wheels were mounted on the axle tubes; this friction would 
undoubtedly be a constant, following Unwin's rule, and the 
total resistance to be overcome by the falling weight is thus 



EFFICIENCY OF WORM GEARING 



69 



H-\-c. We are not much concerned with this constant since it 
includes all losses in addition to the worm losses proper, but 
the point has now been proved that the sliding friction of w^orm 
gear is substantially the same as the sliding friction of a 
journal in its bearing, and we are, therefore, justified in assum- 
ing the same specific value for }j. as in the case of a journal. 
We will, therefore, retain the value of 



^= .002 = tan^ = tan 7' 



(75) 



and presently show that this is correct. 

The losses in a worm gear may be demonstrated graphically 
in the following manner : We may first consider the worm as a 
continuous inclined plane and the teeth of the worm wheel as 
the weight which is to be raised up this plane by the forcing of 
the plane under the weight in a direction parallel to the base of 
the plane. Fig. 31 demonstrates this. 




Fig. 31. 

Taking the ordinary formula for finding the force P if friction 
be nedected P=W ^ =W tan a. It is immaterial whether 



the weight be moved up the plane or the plane slid under the 
weight, since P and p are force and reaction. Taking friction 
into account P=W tan (a + cj)) where </> is the angle of repose 
corresponding to the coefficient of friction. Since it has been 
assumed that the coefficient of friction is .002, it follows that 
the angle of which this is the tangent is 7'. The following 
table shows the values of W for different degrees of the pitch 
angle. 



70 



WORM GEARS 



TABLE VIII 









+ o 


+ o 


S 
^ 






-e- 


S T-T 


« '-* 


d 


O 


-©- 


+ 


^ II 




II 


"-* 




4- 


^ 




Oh 




1—1 


CI 
03 




ti 
^ 


^^ • 


" i 




II 






fin ^ 


^1 


03 


10° 


lO"?! 


.1784 


178.4 


5609. 


.1763 


15° 


15 7 


.2701 


270.1 


3700. 


.2679 


20° 


20 7 


.3662 


366.2 


2732. 


.3639 


25° 


25 7 


.4687 


468.7 


2133. 


.4663 


30° 


30 7 


.5800 


580.0 


1724. 


.5773 


35° 


35 7 


.7032 


703.2 


1422. 


.7000 


40° 


40 7 


.8425 


824.5 


1185. 


.8390 


45° 


45 7 


1.0040 


1004.0 


996. 


1.0000 


50° 


50 7 


1.1966 


1196.6 


835. 


1.1917 


55° 


55 7 


1 . 4343 


1434.3 


698. 


1.4281 


60° 


60 7 


1 . 7402 


1740.2 


575. 


1 . 7320 



The values given in column five would be correct but for the 
fact that they take no account of the work done in the form of 
friction, and are only indicative of the gain in power due to the 
reduction of the worm gearing, they are, however, correct for 
the angle of 45 degrees, as will presently be shown. 

In Fig. 32 the parallelogram represents the developed sur- 
face of the worm, of which A A is the thread. The worm 
wheel is to be revolved in the direction LM. Since the teeth 
of the wheel and the thread of the worm are parallel to one 
another at the point of contact, it follows that the pressure 
between the two is at right angles to the thread surface. Let 
aB represent this force, then this may be resolved into two 
forces La and LB, of which the latter is the useful and the 
former the useless component. If the pitch of the worm is 
increased to CC, the pressure applied along the line gB can be 
resolved into gl and IB, of which LB is the useful component; 
Ba and Bg are by construction equal, and Bl is less than BL, 
hence the force which is available for turning the wheel decreases 
as the pitch is increased and conversely the useless component 
increases with the increasing angle. 



EFFICIENCY OF WORM GEARING 



71 



BLa is a right angle triangle, of which LBa is an angle 
corresponding to the pitch angle. Ba is the force applied. 



BL is the usefnl force. The ratio of these two 



BL 
Ba 



is the cosine 



of the angle LBa. Up to this point, therefore, it would ap- 
pear that, with a constant worm torque, the force available 
for turning the wheel would vary inversely as the cosine of 
the pitch angle. There is, however, another aspect which must 




be considered, and it is the effect of the relative motion between 
the worm and wheel. Thus with a approaching zero value 
the relative motion between the two is very high compared to 
the useful motion of the worm wheel; hence much work is 
dissipated in useless friction. 

Various formulae have been proposed for determining the 
actual mechanical efficiency of worm gears, and so far as 
possible, these will be examined next. 

We will take first, the much quoted formula developed by 
Professor Barr of Glasgow University: 



V = 



tan a (1 —/jl tan a) 
tan oc-\r2/jL 



(76) 



72 WORM GEARS 

For various angles of thread, a, the efficiencies have been 
plotted in column II table IX. 

Next, Professor Unwin (Elements of Machine Design, Vol. 1, 
p. 423) gives the following: 

1 -u cot a (77) 

Column III of the table gives these values. 

Francis W. Davis, M.E., has proposed the following: 

, = !-.( ^ r-) ('«) 

Comparing this equation with (77), it will be noted that 
Unwin's may be written 




//(cot a + tan a) (79) 

l+jj. tan a 



(80) 



COS a sm a + n sm o:^ 

very closely resembling (78). The values of /j. in this equation 
are given in Column IV. 

It must be observed here that neither of the above formulae 
takes into account the pressure angle, and this undoubtedly 
exercises a considerable influence upon the efficiency of the 
gear inasmuch as it very largely governs the normal tooth 
pressure, quite irrespective of the torque. See equation 43, 
Chapter VII. 

The force of friction in the gears is equal to fiXp^, and this 
force tends to resist the gliding of the worm over the wheel 
tooth. It is evident that the path along which the gilding 
occurs is a helical line wound around a cylinder whose diameter 
equals the pitch diameter of the worm, and the length of this 
path is I, the length of thread per revolution measured at the 
pitch line. See equation 15, Chapter IV. The work lost in 
friction equals in foot pounds 

"12" ^^^) 



EFFICIENCY OF WORM GEARING 



73 



The work put into the worm is 

Tndp 
12 
Hence, the efficiency is 



and 

so 



l=Tzd sec a 



/jup^nd sec a\ _ /jup^ sec oc" 



Tnd J ^ \ T , 
Substituting the value already found for p\ we get 



,=1 



jJ-\\ (T cot a tan "9" ) + ( ~ — ) X sec a 



T 



This may be written 



W 



1 — -. ^ \\ 1 +cos^ a tan^-^ 

sm a cos a\ 2 



(82) 

(83) 

(84) 
(43) 

(85) 
(86) 



In the author's opinion, this is the most accurate formula 
of the four examples, and the values are given in column V of 
the table up to the equivalent lead angle of 45 degrees. 

TABLE IX.— EFFICIENCIES OF WORM GEARING FOR VARIOUS 
ANGLES OF LEAD 



I 


II 


III 


IV 


V 


a 
15° 


98.5 


99.2 


-0 
99.2 


99.0 


20° 


98.68 


99.39 


99.38 


99.3 


25° 


98.8 


99.50 


99.45 


99.4 


30° 


99.0 


99.52 


99.54 


99.4 


35° 


99.5 


99.59 


99.56 


99.53 


40° 


99.5 


99.62 


99.59 


99.56 


45° 


99.5 


99.63 


99.60 


99.58 


50° 


99.5 
99.5 


99.63 
99.61 


99.59 
99.55 




55° 




60° 


99.5 


99.58 


99.54 




65° 


99.5 


99.50 


99.45 




70° 


99.3 


99.40 


99.38 









74 WORM GEARS 

It vis clear, therefore, that for all usual lead angles as em- 
ployed in automobile gears, the efficiency is over 99.5 per cent, 
and the conclusion is that the whole question of efficiency is 
one of the simple equivalent of the work put into the worm 
minus that absorbed by friction, and may, for all practical pur- 
poses, be expressed thus for all usual angles 

. = ^^ (87) 

.998 
Numerically, this is ~ — = 99 . 8 per cent. 

Some apology is due to the reader for introducing so simple 
an expression as the outcome of so cumbersome an amount of 
preliminary calculation. The author feels, however, that 
having regard to the high percentage of efficiency this formula 
gives, it would hardly be accepted by engineers unless the 
previous proof were inserted; the demonstration is thus 
rendered complete. 

It may be pointed out that if the table be extended in both 
directions, completing the series of values of a from 0° to 90°, 
it will be found that at both limits the value of tj is zero, which 
is so obvious as to need no elaboration further than to observe 
that if a = 0° or 90°, there will be no movement of the worm 
wheel. 

Practical corroboration of these values of rj has been indis- 
putably furnished by the long series of experiments conducted 
by Messrs. Brown & Sharpe, and published as a memorandum 
by Professor Kennerson in the Transactions of the American 
Society of Mechanical Engineers, 1912, where average values of 
7) for the entire worm gear, including the friction of all the 
bearings, amount to over 97 per cent.,^ and in two instances in 
the series of experiments, the records of which the author has 
been privileged to see, readings were obtained o/ 99 . 8 per cent. 

The efficiencies given in column V of Table IX may, there- 
fore, be accepted as representing what may with care be ob- 
tained in practical working, provided the workmanship is 

^Also corroborated by Nat. Phys. Lab., London in tests on Daimler 
worm-gear, Nov., 1912, (seeaute). 



EFFICIENCY OF WORM GEARING 75 

good, and the mounting and lubrication of the gears properly 
carried out. 

The late Mr. Briggs, of Philadelphia, says in regard to 
friction, '^It is established that for ordinary ratio of wheel to 
worm, say not to exceed 60 or 80 to 1, well-fitted worm gear will 
transmit motion backward through the worm, exhibiting a lower 
coefficient of friction than is found in any other description of 
running machinery.'' 

The selection of .002 as the value of the coefficient of friction, 
earlier in this chapter is thus justified. 

See Appendix B. 



CHAPTER XI 
GENERAL POINTS OF DESIGN OF MOUNTING. 

The design of the worm and wheel have now been fully con- 
sidered, and the stresses set up in the various bearings in- 
vestigated to the point where the designer is in possession of all 
the information to enable him to design the casing in which the 
gears will work. For a worm gear which is to work stationary 
machinery the provision of a suitable casing is a comparatively 
simple matter the only conditions being rigidity and the provi- 
sion of an oil-tight casing to carry the lubricant. The quality 
of rigidity must be insisted upon to ensure the maintenance of 
the correct relative positions of the worm and wheel under the 
heavy stresses to which they are subjected. Weight in such 
cases is of secondary importance and metal need not be spared 
to ensure the desired end. The casing moreover will in such a 
case be bolted securely to a solid foundation which greatly 
facilitates matters. 

Very different are the conditions in an automobile axle where 
weight must be saved at any cost, and strength and lightness 
have both to be studied. 

In the days of the first Lanchester Car, 8 H.P. was all that 
had to be provided for, and the weight of the car was but a few 
hundred pounds. Recent worm gears the author designed are 
capable of transmitting 90 H.P. in a fast pleasure car which 
weighs over three tons with a load of seven passengers and their 
luggage; at the time of writing it is believed that these gears are 
considerably more powerful than any others working in auto- 
mobiles, although their size is by no means excessive. It may 
well be imagined that the provision of a rigid case for these is 
something of a problem; the result has, however, proved 
entirely satisfactory. / 

In the first place there is the question of whether the worm 
shall be placed above or below the wheel. This subject has 

76 



GENERAL POINTS OF DESIGN OF MOUNTING 77 

been discussed by different makers as though it were a matter 
of great moment; in reality it is of no consequence whatever, 
and is purely to be governed by the expediency of other factors 
in the general design. Much more important is the design of 
the axle casing. Based no doubt upon bevel gear practice, the 
casing was originally made in two halves, each containing half 
the bearings required to support the differential, which were 
bolted together around the center, the portion of the case con- 
taining the worm was also in halves, being in fact an extension 
of the main casting. Perfectly satisfactory axles have been 
built on this plan but the difficulties of assembling are very 
great as neither the worm nor the wheel can be seen after they 
are mounted in position. A more modern type is shown in 
Fig. 33 which is an illustration of a rear axle of a five-ton com- 
mercial vehicle, here reproduced by permission of the Fierce- 
Arrow Motor Car Company. It will be seen that the axle 
proper is a pan-shaped casting, with elongations which carry the 
steel tubular extensions on which the road wheels are mounted. 
At the outset therefore a very rigid structure is provided for 
preserving the true alignment of the driving shafts. The worm 
and the wheel are mounted in a single casting, provided with 
strongly ribbed brackets for carrying the bearings of the worm 
wheel, the whole forming the lid of the axle case proper. By 
this arrangement the assembling of the gears and their exact 
adjustment can be carried out on the bench with the worm and 
wheel in full view, giving every facility to the erector for correct 
location in their relative positions; such a method also enables an 
accurate machining operation to be carried out upon the main 
bearings of the gear, and, when examination is necessary, it 
can be made without disturbing any adjustments, the entire 
system of worm wheel, and differential with all their bearings 
being lifted out in one piece for the purpose. 

Much latitude is permissible to the designer, who has a wide 
field of arrangements to select from; it must always be 
remembered that the stresses in the parts supporting the bear- 
ings are high, but with the use of the diagram, Fig. 19, they 
may be determined at a glance, and a selection of suitable 
ball bearings can bie made. 



78 



WORM GEARS 




CO 
CO 

6 



GENERAL POINTS OF DESIGN OF MOUNTING 79 

It may be observed that in the case of a straight worm, that 
is, one having a cyHndrical pitch line, a double thrust bearing 
should be provided at one end (whichever is most convenient) 
the other end can then be left free to expand or contract with 
the difference of temperature which occurs when the worm is 
running. In the case of the hour-glass pattern worm, it is 
difficult to say what happens when it has to expand, presumably 
the casing expands too, and in that case a single thrust bearing 
at either end is the better arrangement. 

Some designers provide a very heavy thrust bearing to take 
the forward drive and a relatively light one for the reverse — • 
it is hard to see any justification for such an arrangement, since 
the reverse gear is almost invariably lower than the first speed 
forward, and the torque at the worm pitch line, and all the 
resultant pressures are in consequence heavier. 

Provision must be made for oil to reach the thrust bearing 
at the rear of the worm when the latter is mounted above the 
wheel. The simplest way, and one which can be confidently 
recommended, is to bore the worm spindle through the center 
and drill small holes, say 3/16-in. diameter, radially into the 
hollow center, these holes being of course drilled between the 
threads of the worm before it is hardened. In the case of a 
large worm for high powered pleasure cars, this method has the 
added advantage of considerably lightening the worm, and the 
hollow in the spindle carries the oil back to the thrust bearing in 
a steady stream. 

Provided precautions are taken to prevent the oil from 
leaking out through the arms of the axle, it will be found that 
the same oil may be used almost indefinitely — at any rate 
from 5000 to 7000 miles, and it may be observed that the oil 
level should be somewhat below the center of the casing. A 
small vent pipe may be provided if thought necessary, to allow 
for expansion of the air when the gears get warm, this will 
prevent the oil being forced out round the axles by any pres- 
sure so formed; such a pipe must, however, be closed by a cap 
drilled with a very small hole, say 1/16-in. diameter and the 
pipe should be loosely plugged with cotton to prevent the 
entrance of dust. 



80 WORM GEARS 

The author has intentionally refrained from elaborating to 
any extent upon the details of the casing, which are best left 
to the judgment of the designer; sufficient attention has, 
however, been drawn to the more important features to enable 
a satisfactory design to be produced if the principles set out in 
the earlier chapters of this book are closely followed. In con- 
clusion, let it be pointed out that no workmanship can be too 
accurate in the manufacture of worm gearing, and unless 
facilities exist for this, a satisfactory result cannot be looked for. 
If, on the other hand, proper precautions are taken, a worm gear, 
correctly designed and mounted, is, as Mr. Briggs has observed, 
probably the most efficient piece of machinery known; and its 
efficiency is only equalled by its durability and silence of 
operation. 



CHAPTER XII 



RECAPITULATION OF FORMULiE USED 

^ .. ^ Wheel circumference 

Gear ratio G = , — -^ (1) 



worm lead 

L 

L 



c-f tf) 



J 



Measurement of 

(a) Axial pitch =P 

(6) Normal pitch P' = P cos a [ (2) 

(c) Circumferential pitch P" = P cot a 
Circumference of worm = 7id = nP cot a (3) 

Ttd 
Number of threads on worm = n = -=, — - — - (4) 

r* cot a ^ 

Lead = L= -^ (6) 



G 

Lead angle = a 



Length of worm=gf 



L^Pn (14) 



tan a = — , (7) 

ltd ^ 

tan a = ^ (8) 

Pn 

tan a =^ (17) 



g= 2iS"sm| (9) 

SI 



82 WORM GEARS 

Number of teeth in contact = n' 







n' = 


2R'' sin ^ 
P 


Centers of worm and wheel = c 




D + d 
'= 2 


Length of thread per 


revolution of worm 




= l = 7[d sec a 


Rubbing velocity of worm = 'i; 



(11) 



(12) 



(13) 



(15) 



Tzd sec ap . 

v = — ^^ — it. per second Uo) 

Ratio between axial pressure angle d and normal pressure 
angle ¥ 

n tan^^- 

tan -- = (19) 

2 cos a 

we 

tan— = tan ~ cos a (20) 

Normal tooth pressure = p' 

T 

^'^sin(a + ^) ^^^^ 

V'=~~^, (26) 



cos X 



rp 



p'=J(rcot«tan|)=+(^j-^)^ (43) 



RECAPITULATION OF FORMULA USED 83 

Resultant, normal pressure angle = z 

¥ 

tan 2= cos a tan- (40) 

Tangential tooth pressure (due to tractive resistance) 

= P=^ (28) 

Length of wheel tooth =ii 

Safe load on wheel tooth = x 

x = 2,2 l^P (34) 

Worm end thrust = p 

p = Tcota (49) 

Separating force between worm and wheel =/ 

/= T cot a tan - (53) 



Radial thrust on worm bearings = S 



S = ^l (7^cotatan|^V-+ r (56) 



Subtended angle of worm = /? 



f 



^ = 2JJP^ ^''^ 

360 g tan a 
^ n {d+.63m P) ^ ^ 

Tooth pressure velocity constant = ^'1; 

p'v^ 17000 (67) 

Heat generated in gears = H 

778 ^ ^ 



84 WORM GEARS 

Area of axle surface required to dissipate heat = a 

a = .635 sq. ft. (73) 

Coeflficient of friction = /i 

// = . 002 = tan a = tan 7' (75) 

Efficiency (accurate) = t^ 



'^ = 1 - sin jWaV^ +(ios'c, tan^ ~ (86) 



Efficiency (approximate) 



V- ^ (87) 



APPENDIX A 

ALTERNATIVE METHOD OF CALCULATING STRESSES 
IN WORM GEARING 

On page 25, a value of 60° is assumed for the axial 
included thread angle d. The relationship between the 
axial included angle and the normal angle of the thread is 
shown in equation 20. In Fig. 19, page 45, the value of this 
angle is shown in the form of a curve for different lead angles 
assuming a constant value of 60° for the axial thread angle. 
A study of Fig. 19 shows that a different milling cutter 
would be required for every different value of the lead angle 
in order to produce a constant axial thread angle of 60°. 
For considerations of manufacturing, it is far easier and 
more economical to keep the normal thread angle constant 
Sit 60° by which means all the cutters would be ground to 
the same angle and the axial thread angle would be allowed 
to take care of itself. Moreover, although slightly in- 
creased by this method, the tooth pressures are not sensibly 
affected and the calculation becomes a much more simple 
matter. In the following figure (a), let ab be the axis of 
the worm; and c, d, e, /, the surface of the pitch line cylinder 
of the worm opened out flat, one thread of the worm being 
shown. Let Tx represent the magnitude of the torque; 
px will represent the normal pressure, and for a square 
thread, the magnitude of this will be expressed by 

^^ = .^^ f ^ ■ N (25, Chapter V) 
sin {a + (p) 

(p can be omitted as being too small to make any practical 
difference. Let plane c, d, e, f, be viewed in perspective, as 
shown in the following figure (h). 

Draw the perpendicular pq and make the angle pxq 

85 



86 



APPENDIX 



equal to 30°, that is to say, equal to the pressure angle — • 

Si 

Then the triangle pxg can be solved for xq by the following 
equation 

xq = "px sec 30 



T 



px = — 



Therefore, 



xq = 



sin a 

T sec 30 

sin a 



and as sec 30° is a constant, the equation can be written 

1.15477^ 

xq = — ; = V 

sm a X 



(a) 




The following table gives the value of xq, the normal 
tooth pressure for various lead angles. 



TABLE X 



Lead angle 


Tooth pressure. {T = 


1 lb.) 


15° 


4.46 




20° 


3.375 




25° 


2.73 




30° 


2.31 




35° 


2.01 




40° 


1.795 




45° 


1.630 





APPENDIX 87 

The other stresses shown in the diagram, Fig. 19, the 
end thrust of the worm, radial thrust of the worm, side 
thrust of the worm, and the separating forces are not 
affected. 

Tooth pressure for 30° pressure angle may be written 

T = ^M^ inch lbs. 

n 

d = pitch diameter 
.5d = pitch radius 

As torque varies inversely with .5d, we get 
63,024P _ 126,048P 



T Sit pitch line = 



n X .5c^ nd 

, 1.15477^ 



and p 

sm a 

and substituting 

, 1 1 ..7 ^ 126,048 X P 145,447P 

p' = 1.1547 X b-^ " = — J—- 

nd sin a nd sm a 



APPENDIX B 

EFFICIENCY OF WORM GEARING 

Equation (84) gives the following value for the efficiency 
-, ( iiv' sec a\ 

Substituting for p' the value obtained (from Appendix A), 
we get the following equation 

/ T 

U X 1.1547 ^ X sec a 

1 sm a 

, = 1 _ ^ _ — 

This may be written 

.002 X 1.1547 X ~— X sec a 
-, , sm a 



Or more simply 

77 = 1 - f .0023094 ^-~^-^) 
\ sm a/ 



This can, of course, be expressed empirically for any 
pressure angle as follows: 

/ fx sec — sec a 

, = 1 _ I — 4 — 

\ sm a 

The efficiencies calculated by this formula are given 
in the second column of the following table: 

89 



90 



APPENDIX 

TABLE XI 



Lead angle 


Efficiency 


15** 


99.076 


20° 


99.28 


25" 


99.39 


30° 


99.46 


35° 


99.508 


40° 


99.53 


45° 


99.539 


50° 


99.53 



Basing his reasoning upon the relative velocity of the 
worm and the worm wheel, P. M. Heldt gives the following 
formula for efficiency: 



V = 



cos ^ — M tan a 

Jit 

cos ^ + M cot OL 



It will be found that this formula gives precisely similar 
results to those given in the above table, assuming in 
both cases, at 30° pressure angle. 



APPENDIX C 

REVERSIBILITY 

There is a very erroneous opinion that the reverse 
efficiency of any worm gear is considerably lower than the 
forward; that is to say that the efficiency is higher when the 
worm is driving the wheel than it is when the worm wheel 
drives the worm. 

This misconception as to reverse efficiency, frequently 
referred to by writers on worm gearing, is due to the mistake 
almost invariably made in over-estimating the value of 
the coefficient of friction. From various experiments it 
has been substantiated that this value does not exceed 
.002 and is sometimes lower. 

Again the pressure angle is cited by some as having a 
great effect in reversibility. As a matter of fact it will now 
be shown' that the reverse efficiency is within six thou- 
sandths of 1 per cent, of the forward efficiency. 

We will take the case first of a square thread, ^.e., one 
which has no pressure angle, and assume a pitch line torque 
of 1 lb. both forward and reverse (see Fig. 7, page 27). 

T 

Tooth pressure p' = ^ and T = 1. 

sm a -{- (p 

so p' will be 



sin a -\- (p 
neglect (p. * ■ 

Then for the forward angles we have the following 
backward angles and other functions as follows*: 

91 



92 



APPENDIX 

TABLE XII 



Angles 


Sin a 


Sinf 


Tooth pressures 


Forward 
= a 


Backward 


Forward 

1 
Sin a 


Backward 

1 
Sinr 


15 
20 
25 

3o; 

35 
40 
45 
50 


75 
70 
65 
60 
55 
50 
45 
40 


0.2588 
0.3420 
. 4226 
0.5000 
0.5735 
0.6427 
0.7071 
0.7660 


0.9659 
0.9396 
0.9063 
0.8660 
0.8191 
0.7660 
0.7071 
0.6427 


3.87 

2.92 

2.37 

2.00 

1.745 

1.556 

1.414 

1.305 


1.035 
1.065 
1.104 
1.155 
1.220 
1.323 
1.415 
1.555 



fjL sec ^sec a 
The efficiency is 77 = 1 — -r^ and since pressure 



sm a 



^ 



angle = 0, the sec — becomes 1, hence for a square thread 



1 - 



M sec a 

sin a. 



TABLE XIII 



Angles 


Sin 


Sec 


Sec/Sin ^ 


Efficiency, 
per cent. 




15 




0.2588 


1.0352 


4.00 XO 


.002=0.00800 


99.200 


'S 


20 




0.3420 


1.0641 


3.11 


0.00622 


99 . 378 


03 



25 




0.4226 


1.1033 


2.61 


0.00522 


99 . 478 


30 




0.5000 


1 . 1547 


2.308 


0.004616 


99.5494 


^ 


35 




0.5735 


1.2207 


2.13 


0.00426 


99.574 




40 




0.6427 


1.3054 


2.03 


0.00406 


99.594 


45] 




0.7071 


1.4142 


2.00 


0.00400 


99 . 600 


50 


"xi 


0.7660 


1.5557 


2.03 


0.00406 


99.594 


55 


03 


0.8191 


1.7434 


2.13 


0.00426 


99.574 


60 


^ 
r ^ 


0.8660 


2.0000 


2.308 


0.004616 


99 . 5494 


65 



03 


0.9063 


2.3662 


2.61 


0.00522 


99 . 478 


70 


pq 


0.9396 


2 . 9238 


3.11 


0.00622 


99 . 378 


75 J 




0.9659 


3 . 8637 


4.00 


0.00800 


99.200 



APPENDIX 



93 



Next take the case of the square thread but including the 
effect of friction (the value of ^). 

We have at the following angles the values shown : 



TABLE XIV 



a 


a+ip 


Sin 


Sec 


Sec/Sin 


Efficiency, 
per cent. 


15 


15° 7' 


0.2607 


1 . 0358 


3 


97 : 


X0.002 = 


=0.00794 


99 . 206 


30 


30° 7' 


0.5071 


1.1560 


2 


280 




0.00456 


99.544 


45 


45° 7' 


. 7085 


1.4171 


2 


000 




0.00400 


99.600 


60 


60° 7' 


0.8670 


2 . 0070 


2 


310 




. 00462 


99.538 


75 


75° 7' 


0.9664 


3.8932 


4 


03 




. 00806 


99.194 



This may for convenience be written: 



TABLE XV 





Efficiency, per cent. 


Difference, 


Ol 


Forward 


Backward 


per cent. 


15 
30 
45 


99.206 
99.544 
99.600 


99.194 
99 . 538 
99 . 600 


0.008 
0.006 
. 000 



Lastly we will take into account both the friction and the 
pressure angle, assuming the usual value of 30° for this. 
The efficiency is expressed as 



7/ = 1 —IfJ, 



sec 30 sec a 



sm a 



or 77 = 1 - ( 0.0023094 



sec a 
sin a 



Taking the values as in column 5, Table XIV, we obtain: 



94 



APPENDIX 



TABLE XVI 



a 


Sec/Sin 


Efficiency, per cent. 


15° 


3.97X0.0023094=0.0091683 


99.08317 


30° 


2.28 


0.0052655 


99.47345 


45° 


2.00 


0.0046188 


99.53812 


60° 


2.31 


0.0053347 


99.46653 


75° 


4.03 


0.0093069 


99.06931 



which may be written: 



TABLE XVII 





Efficiency, per cent. 


Difference, 


oc 


Forward 


Backward 


per cent. 


15 
30 
45 


99.083 
99.473 
99 . 538 


99.069 
99.466 
99 . 538 


0.014 
0.007 
0.000 



INDEX 



Addendum, 10, 16 
Annealing worms, 7 
Area of physical contact, 50, 52 
Axial pitch of worm, 10 

Bach and Roser, 60, 64 
^ value of, 48 

^ variations of, author's experi- 
ments, 52 
Bronze alloy for worm wheel, 5 
Brown & Sharpe, efficiency experi- 
ments, 74 
Bruce, Robert A., deductions, 50 

Case hardening, worms, 6 

Casing, design of, 76 

Centers of axes, 19 

Circular pitch of wheel, definition of, 

10 
Circumferential pitch, definition of, 

10 

Daimler axle, test of, 33, 74 
Dedendum, 10, 16 
Dennis, early worm gear, 2 

Efficiency, Daimler worm axle, 74, 
foot note, 
formula for, 72, 73, 74 
of worm gearing, 63 et seq., 89 

Forms of teeth, 29 
Friction, Beauchamp Tower experi- 
ments, 63 
' coefficient of, 63 
in axle, author's experiment, 67 

Gear ratio, calculations for, 17 

definition of, 8 
Gliding angle, 26 
Globoid gear, 3 

Hardness of materials used, 5, 6^ 



Heat generated, 58 
Hindley worm gear, 7, 8, 29 
Hobbing, 6 

Included angle, calculations of, 25 

of thT-ead, definition of, 11 
Interference of wheel teeth, 22 

Kennerson, Professor, on Brown & 
Sharpe experiments, 74 

Lanchester, early worm gear, 2, 8 

gear, 30 
Lead angle, calculations for, 17, 19 
definition of, 10 

definition of, 9 

of worm, 19 
Length of teeth, calculation of, 27 

of thread, 19 

of worm, definition of, 10 
determination of, 17 
Lost work of worm, 21 
Lubricating oil, 57 

Manufacturing methods, 6, 7 
Materials for worm gears, 5, 6 
Mechanical efficiency, 72, 73, 74 

Normal pitch, definition of, 10 

Oil, film, 56 

lubricating, 57 

Pierce-Arrow Motor Car Co. worm 

axles, 2, 77 
Pitch, calculation of, 17 

in relation to horsepower, 14 
line of wheel, definition of, 9 
of worm, definition of, 10 
selection of, 13 
Pressure angle, 22 
definition of, 11 



95 



96 



INDEX 



Proportions of gears. 

Rack, 8 

Radiation of heat from axle, 59 
Reversibility, 8, 23, 89 
Rubbing velocity, 19 

Screw, definition of, 1 
Strength of bronze alloy, 5 

of steel for worm, 6 

of teeth, 34 
Stresses in gears, 38-46, 85 
Subtended angle, definition of, 11 
Symbols, 11, 12 

Temperature, coefficient, 58 
maximum permissible, 58 
of gears, 51 

Teeth in contact, number of, 17 



Thread, form of, 22 

proportions of, 26 
Threads, number of, 17 
Tooth pressure, effect of varying, 62 

pressures, calculations of, 28 
Tower, Beauchamp, friction experi- 
ments, 63, 68 
Tractive resistance, 34 

Velocity, rubbing, effect of varying, 

62 
Viscosity of lubricant, 52 

Width of worm wheel, 47 
Worm gearing, definition of, 8 

location of, 77 

material for, 6 

wheel, material for, 5 
size, determination, 16 



/ 



